Can a Catholic have communion if they married outside of the Catholic Church?

I think this question can be answered without any reference to marriage convalidation. In fact, I think they are unrelated issues.

Guideline of when can Catholics receive communion is given in this document, prepared by the Diocese of Westminster. These are:

1. you must be in a state of grace. This mean you have not committed a mortal sin. About this, the document states:

   A mortal sin is any sin whose matter is grave and which has been committed wilfully and with knowledge of its seriousness. Grave matter includes, but is not limited to, murder, receiving or participating in an abortion, h*m*sexual acts, having sexual intercourse outside of marriage or in an invalid marriage, and deliberately engaging in impure thoughts (Matt. 5:28–29). Scripture contains lists of mortal sins (for example, 1 Cor. 6:9–10 and Gal. 5:19–21). For further information on what constitutes a mortal sin, see the Catechism of the Catholic Church.

2. you must have been to confession since your last mortal sin.

3. you must believe in the doctrine of transubstantiation.

4. you must observe the Eucharistic fast (no food or non-water drinks one hour before communion).

5. one must not be under an ecclesiastical censure. On this respect the Canon Law states:

"Those who are excommunicated or interdicted after the imposition or declaration of the penalty and others who obstinately persist in manifest grave sin are not to be admitted to Holy Communion" (CIC 915).

Now, key here is, in my view, 1 and 5 (the others could be said to be standard among Catholics, regardless of the type of marriage they have). This is, was your marriage with a non-Catholic, by definition, an act of mortal sin? Also, has such marriage, by definition, derived in a form of ecclesiastical censure? The answer to both of them is no. The answer is partly in this article explaining the Canon Law, commenting the state of Catholics in non-sacramental marriages. The key here is the fact that

the Catholic Church does indeed permit Catholics to marry non-Catholics, whether baptized or not

and

Note that there is nothing necessarily wrong with a Catholic having a non-sacramental marriage. If the Catholic party arranged with his/her pastor to obtain a dispensation in advance, and was married in a Catholic wedding ceremony, the marriage is recognized by the Catholic Church as valid. In no way can a Catholic who was married in this sort of scenario be faulted for doing something morally objectionable!

This other article from the same website states:

as the Second Vatican Council’s Document on Religious Freedom Dignitatis Humanae stated unequivocally,

This Vatican Council declares that the human person has a right to religious freedom. This freedom means that all men are to be immune from coercion… in such wise that no one is to be forced to act in a manner contrary to his own beliefs… (DH 2 ).

Within the context of marriage, this means that the Catholic Church will not—because it cannot—claim that you can only get married if you are a Catholic, or only if you’re a baptized Christian.

In conclussion, having married a non-Catholic does not mean by itself that you have committed a sin that invalidates you from receiving the Holy Communion, nor has by itself being a reason to receive an ecclesiastical censure. Thus, as long as you, as any other Catholic, obey rules 1 to 4, you can receive the Holy Communion.

About receiving communion currently

Based on this guidance from the Catechism, unless you've been adhering to the annual "Easter Duty" you may need to go to confession first before you receive communion. (It may be that simple).

CC 1389 The Church obliges the faithful to take part in the Divine Liturgy on Sundays and feast days and, prepared by the sacrament of Reconciliation{aka confession} to receive the Eucharist at least once a year, if possible during the Easter season. But the Church strongly encourages the faithful to receive the holy Eucharist on Sundays and feast days, or more often still, even daily.

About your marriage

You may be seeking to convalidate your marriage. Based on what you've presented in your question, there aren't enough details to assess whether or not the Church would find it to be sacramental. (Canon Law; 1060 Marriage possesses the favor of law; therefore, in a case of doubt, the validity of a marriage must be upheld until the contrary is proven).

How to convalidate the marriage

1. Meet with your local Catholic pastor, explain the situation you find yourselves in, and the details of your marriage, and he will be able to advise you on the path to take for your particular situation. Details matter.

   In Catholic canon law, a validation of marriage or convalidation of marriage is the validation of a Catholic putative marriage. A putative marriage is one when at least one party to the marriage wrongly believes it to be valid.

2. Convalidation is not "getting married a second time" but it is a method by which the Church officially confirms that the marriage is indeed sacramental. (In my experience, there was usually a rite or ceremony or celebration of varying complexity depending on the wishes of the couple). The difference between a licit and a valid marriage may seem obscure, but from the point of view of adhering to Canon Law it matters to the Church. (The Church, to affirm any marriage, has to follow its own rules).

   Based on the Catholic Canon Law a valid marriage requires certain elements:

   THE FORM OF THE CELEBRATION OF MARRIAGE
   Can. 1108 §1. Only those marriages are valid which are contracted before the local ordinary, pastor, or a priest or deacon delegated by either of them, who assist, and before two witnesses according to the rules expressed in the following canons and without prejudice to the exceptions mentioned in cann. ⇒ 144, ⇒ 1112, §1, ⇒ 1116, and ⇒ 1127, §§1-2.
   Can. 1117 The form established above must be observed if at least one of the parties contracting marriage was baptized in the Catholic Church or received into it and has not defected from it by a formal act, without prejudice to the prescripts of ⇒ can. 1127, §2.

   The most likely finding by the pastor, since you are the Catholic party, would be technically termed "a defect of form" since you joined in matrimony under other circumstances than are called for by Canon Law. That doesn't mean that your bond is not sacramental, particularly as you are both baptized, but the Church has to look into all particulars to confirm that.

3. You may not be able to receive communion until convalidation, but you may, since the details of your situation matter(see the above about guidance on communion). Your situation needs to be laid out to the pastor and discussed in detail so that you can get the advice that suits your situation.

   If you'd like to take it a step higher, go to your local diocese at the Chancery. They can advise you on how your marriage can be convalidated in your diocese. There may be other reasons for the Church to find the marriage invalid¹

If your case is one where your consent upholds as it did on the day you took your vows, but the "defect of form" is found(as above), and you are still married, you may be eligible for a radical sanation

Can. 1161 §1. The radical sanation of an invalid marriage is its convalidation without the renewal of consent, which is granted by competent authority and entails the dispensation from an impediment, if there is one, and from canonical form, if it was not observed, and the retroactivity of canonical effects. {My deacon used to call this "short form" convalidation}

§2. Convalidation occurs at the moment of the granting of the favor. Retroactivity, however, is understood to extend to the moment of the celebration of the marriage unless other provision is expressly made.

§3. A radical sanation is not to be granted unless it is probable that the parties wish to persevere in conjugal life.

Can. 1163 §1. A marriage which is invalid because of an impediment or a defect of legitimate form can be sanated provided that the consent of each party perseveres.

We have a number of questions and answers on convalidation, to include these two here.
3 and 2.

If you search for the tags marriage, Catholicism and convalidate (or convalidation) you'll find a number of other questions (with answers) similar to yours.

One of the things I learned in the RCIA ministry is that in the case of marriage between two baptized persons, if one of them is Catholic then the Catholic Church expects the Catholic party to adhere to Canon Law, but it does not require that the other party become Catholic for marriage. Since you have already married outside of the Catholic Church, it can get a bit more complicated, but it may be very simple (and accepted as sacramental) if neither of you was ever married previously. So once again, sit down with a priest and discuss the details.

Can. 1059 Even if only one party is Catholic, the marriage of Catholics is governed not only by divine law but also by canon law, without prejudice to the competence of civil authority concerning the merely civil effects of the same marriage.

Can. 1060 Marriage possesses the favor of law; therefore, in a case of doubt, the validity of a marriage must be upheld until the contrary is proven.

Can. 1107 Even if a marriage was entered into invalidly by reason of an impediment or a defect of form, the consent given is presumed to persist until its revocation is established.

¹ MIXED MARRIAGES

Can. 1124 Without express permission of the competent authority, a marriage is prohibited between two baptized persons of whom one is baptized in the Catholic Church or received into it after baptism and has not defected from it by a formal act and the other of whom is enrolled in a Church or ecclesial community not in full communion with the Catholic Church.

Can. 1125 The local ordinary can grant a permission of this kind if there is a just and reasonable cause. He is not to grant it unless the following conditions have been fulfilled:

1/ the Catholic party is to declare that he or she is prepared to remove dangers of defecting from the faith and is to make a sincere promise to do all in his or her power so that all offspring are baptized and brought up in the Catholic Church;

2/ the other party is to be informed at an appropriate time about the promises which the Catholic party is to make, in such a way that it is certain that he or she is truly aware of the promise and obligation of the Catholic party;

3/ both parties are to be instructed about the purposes and essential properties of marriage which neither of the contracting parties is to exclude.