Does the Holy Spirit's procession from the Father and Son infringe on the co-equality of the Trinity?

The short answer would be yes.

When speaking of the persons of the trinity we can either speak of an attribute in relation to one person (hypostatic properties, which is how we differentiate the persons), or to all of the persons, which would be the attributes, actions or energies of God.

In the case of the father his hypostatic property is that he is the sole arche or sole cause of the other persons of the Trinity, also this is what Athanasius and the Cappadocians teach so quoting them while they are talking about the economic procession and trying to attribute that to hypostatic procession is useless.

Therefore the son can't also have this attribute of causation or else the spirit would also have this attribute and we would have infinite divine persons or we would have to say the father and the son both have an ability that the spirit doesn't share in so the ability wouldn't come from the shared divine essence or the spirit would have it, causing what I stated above. The only other solution here would be that causation is a hypostatic property of both the father and son, therefore making them the same person.

So if we hold to the doctrine of the double hypostatic procession of the spirit then we find that

1 - the son is the father

2 - we have infinite divine persons

This is why people like St. Gregory Palamas and even some early popes like Leo III rejected the filioque.

Short answer: No

The scriptures describe the Holy Spirit as both the "Spirit of [the] Father" (Matthew 10:20) and the "Spirit of Christ" (Romans 8:9, 1 Peter 1:11) / "Spirit of Jesus" (Acts 16:7) / "Spirit of Jesus Christ" (Philippians 1:9). From this, we deduce that the relations of both the Father and the Son to the Holy Spirit are accurately described in Geremia's answer by the term "spiration". The objections advanced against the use of the term "procession" could equally be applied to its reverse relationship of "spiration" - that they should not be so applied should be inferred from the subtext of your question that the co-equality of the Trinity is indeed correct doctrine (which I will omit arguing the case for as it doesn't seem to be at issue).

Furthermore, as the answers to this previous question indicate, there are biblical proof texts* for the concept of procession from the Son, whereas the contrary argument relies merely on sophistic reasoning and not the authority of scripture. I would add, in addition to the scriptures quoted by James Black's answer to that question, that consideration of the differing roles played by the persons of the Trinity in the act of Creation would bring us to the same position as:

For by him [that is the Son] all things were created... - Colossians 1:16 ESV

and yet the Holy Spirit was intimately involved in the act of Creation:

...And the Spirit of God was hovering over the face of the waters. Genesis 1:2 ESV

Elsewhere, the nature of the procession of the Holy Spirit is described as from the Father and through the Son. It seems any quibble with the word stems from differences in understanding its meaning and usage.

Any problem that the filoque clause presents to the co-equality of the Trinity is miniscule in comparison to that posed by those who oppose it and maintain instead the doctrine of the "Monarchy of the Father" which is a form of Subordinationism; for although the doctrine the filioque represents (procession of the Holy Spirit from the father and the son) was taught by many of the Church Fathers (the wikipedia article lists Tertullian, Jerome, Ambrose, Augustine, Athanasius and Cyril of Alexandria amongst others), the most likely reason for it's first official use in creedal form was to bolster a Trinitarian Christology in order to counter the continued threat (particularly in the West) of Arianism (the declared heretical extreme of Subordinationism) to the Nicene faith (source).

*From the wikipedia article on the filiquoe:

In John 16:13-15 Jesus says of the Holy Spirit "he will take what is mine and declare it to you", and it is argued that in the relations between the Persons of the Trinity one Person cannot "take" or "receive" (λήμψεται) anything from either of the others except by way of procession.14Texts such as John 20:22 ("He breathed on them and said: Receive the Holy Spirit"), were seen by Fathers of the Church, especially Athanasius, Cyril of Alexandria and Epiphanius of Cyprus as grounds for saying that the Spirit "proceeds substantially from both" the Father and the Son.15Other texts that have been used include Galatians 4:6, Romans 8:9, Philippians 1:19, where the Holy Spirit is called "the Spirit of the Son", "the Spirit of Christ", "the Spirit of Jesus Christ", and texts in the Gospel of John on the sending of the Holy Spirit by Jesus (14:16, 15:26, 16:7).14(source)

There are four relations in the Trinity:

1. paternity (the relation of the Father to the Son)

2. filiation (the relation of the Son to the Father)

3. active spiration of the Holy Spirit (the relation of the Father and the Son in respect to the Holy Spirit)

4. passive spiration of the Holy Spirit (the relation of the Holy Spirit in respect to the Father and the Son)

The Holy Spirit lacks nothing of the Father and the Son. Just as there is a relation of the Father and Son to the Holy Spirit, called active spiration, so there is also a relation of the Holy Spirit to the Father and the Son, called passive spiration.

The green lines are the relations:

(image source)

Notice that for each Person, there are always two "from" relations and two "toward" relations. Thus, no Person has fewer relations than the others have, and it's not like one type of relation is somehow greater than another. Thus, all Persons are co-equal.

Also, the divine relations are God's essence.