Will God give us the gift of vision, dream, and prophecy if we request Him?

First a point of clarifications:

In the widest sense, pentecostalism is the belief that there is a second, post-regeneration experience called the baptism with the Holy Ghost usually through the laying on of hands, which is externally manifested often with speaking in tongues (but this is not a universal belief). Charismatic Christianity believes in the active working of the supernatural gifts of the Holy Ghost in the normal life of the church such as miracles, healings, dreams, visions, prophecy, and speaking and interpreting tongues.

Pentecostals would be Charismatic Christians, but Charismatic Christians might not be Pentecostals. Assemblies of God is an example of the former, and I think theologian Gordon Fee might be an example of the latter.

To answer your question, it is the standard teaching in the Charismatic Movement that God still speaks to individuals with prophetic gifts: prophecy, visions, words of knowledge and wisdom. Prophetic Gifts are not considered to be "foretelling the future" but rather "words from God for edification". There is an emphasis on the personal relationship with God through his presence and Spirit, which manifest in such experiences.

Usually Pentecostal and Charismatic churches are MUCH less credal than Reformed and Catholic denominations because it emphasises the personal relationship so much, which means there is a MUCH wider diversity of beliefs and applications of what is believed. In the more conservative charismatic churches, there is a strong emphasis on "testing every spirit" and therefore a "healthy" scepticism.

The Old Apostolic Church with roots in Catholicism was founded in the 19th century and is one of the oldest Pentecostal movements and (at least here in South Africa) make a very big deal about God speaking to individuals regularly through dreams and consulting the "priest" to help with interpretation.

So to answer your questions, it is hardly meaningful to say standard Pentecostal teaching, but what might be considered standard Pentecostal and Charismatic teaching is that God still speaks to individuals about personal matters through the prophetic gifts of the Holy Ghost: Dreams, visions, prophecies, interpreted tongues, and maybe some other; and "Spirit-filled" believers can and should experience it.

Can I ask for God to speak to me through these gifts?

It is also important to note that Pentecostals (and most Charismatics) teach that there is also a difference between charismatic gifts (the focus of 1 Cor 12&14) and Offices (Eph 4:11 & 1 Cor 12:28). They mostly believe all believers can operate in charismatic gifts, like prophecy, visions, tongues, even miracles and healings. But some are "set" in the church to fulfil a certain function in ministry:

And God hath set some in the church, first apostles, secondarily prophets, thirdly teachers, after that miracles, then gifts of healings, helps, governments, diversities of tongues. (1Co 12:28)

These offices would have a function and role in the church, usually, focussing on equipping all believers to operate in the Charismatic gifts and leading the church with their "primary gift" for the purpose of growing into the unity of faith (Eph 4:13)

With this in mind, Biblically (if you grant the continuation and distinction of Charismatic gifts operating and Certain Offices) we can say the following:

1. Gifts are distributed by the Holy Ghost to ANYONE according to His will (1 Cor 12:11)
2. All offices are appointments by Jesus for the Church to grow into the unity of Faith (Eph 4:10-13)
3. The gifts and offices' primary and best purpose is to edify the body not the individual (1 Cor 12:7; 14:1-4, Eph 4:12)
4. If it is from God it will and must be administered orderly (1 Cor 14:33)
5. Prophetic words must be subject to the Scriptures and prophetic Office (1 Cor 14:32)
6. Therefore Prophetic words must be judged/interpreted (1 Cor 14:29, 1 Thess 5:20-21)
7. The testimony of Jesus is the spirit of prophecy.(Rev 19:10)
8. The Good gifts are given to those who "are hungry" and ask for it (Luke 11:13)

So to answer your question, it will appear that if you grant the continuation of Spiritual Gifts, anyone can and should absolutely ask for it, but its purpose, like anything we ask God for, is not to consume it on your own lusts (James 4:3), but rather on the edifying of the Body. As a member of that body personal ministry to you by the Holy Ghost will obviously benefit the body, but not if it only serves to promote your own self-interest like your career or relationships apart or instead of the Testimony of Jesus.

I think most will say that to desire an Office would also be a good thing (similar to 1 Tim 3:16) but that it is ordination by Jesus Christ not for all. Many believe that some of these offices closed with the close of the Bible Canon, especially the Office of Apostle but some would include the Prophetic Office also.

Most Charismatics would use 1 Cor 14:1 and 1 Thess 5:19-21 to encourage everyone to operate in spiritual gifts and Luke 11:13 to receive it by asking.

Follow after charity, and desire spiritual gifts, but rather that ye may prophesy. (1Co 14:1)

Quench not the Spirit. Despise not prophesyings. Prove all things; hold fast that which is good. (1Th 5:19-21)

If ye then, being evil, know how to give good gifts unto your children: how much more shall your heavenly Father give the Holy Spirit to them that ask him? (Luk 11:13)

But in practice, the emphasis on the personal relationship with God the Holy Ghost, means that it is often applied primarily personally to know God's will for your own life. The idea of "putting out a fleece" like Gideon (in my experience as a Pentecostal) is very common among Charismatics to test if something is from God, while submitting your prophetic word or dream for judgement is very uncommon.

That being said, regardless of the continuation of any gifts or offices, like the Lord Jesus Christ, the Holy Ghost is God and can speak independently of any gift or office. In which case, there is no judging just obeying. This seems to be the Apostles' experience in Acts 8:29; Acts 10:19; Acts 13:2 and Acts 16:6-7. and certainly the Old Testament experience for the Prophets that spoke moved by the Holy Ghost - it was God speaking, not through a Gift or an Office, but rather in his own Devine right. Faithful Pentecostal expositors are unlikely to condone the use of these as examples of how the Gifts of the Holy Ghost or active offices administer the prophetic word of God today.