The letters of Paul were already being passed around by that time. Christians may have had access to a written gospel as well, ( if not then the gospel would have been passed down orally, as many were illiterate anyway).

Many christan writings are dated to the period. Here is a list with dates and links to the writings: http://www.earlychristianwritings.com/

100 AD was long before the invention of the printing press; copying a written document was quite labor-intensive at that time. Individually owned copies of sizable written documents would have been uncommon.

On the Jewish side of things, a synagogue might have had copies of various writings. For example, Luke 4:16-20 has Jesus reading from a synagogue's copy of the book of Isaiah.

I don't know how common it was for christian groups at that time to have copies of written documents. But christian tradition and practice was certainly influenced by Jewish tradition and practice -- which is suggestive.

If we change "possess" to "have access to," the answer is "yes." We have several non-biblical Christian writings from that time, and they quote liberally from what we would now call the New Testament.

Take for example the letter of Clement of Rome to the Corinthians:

Take up the letter of the blessed Apostle Paul. What did he write to you at the time when the Gospel first began to be preached? Truly, under the inspiration of the Spirit, he wrote to you concerning himself, and Cephas, and Apollos, because even then parties had been formed among you. But that inclination for one above another entailed less guilt upon you, inasmuch as your partialities were then shown towards apostles, already of high reputation, and towards a man whom they had approved.

Here, it is taken as a given not only that Corinth keeps and treasures Paul's epistle to them, but that the church in Rome has a copy as well. The situation is reversed in the previous paragraph when Clement quotes Romans 12:5, implying that the Corinthians would likewise have a copy of Paul's epistle to the Romans. In that same paragraph, he also quotes Jesus (in this case, a saying found in all the synoptic gospels).

On his way to be executed, Ignatius of Antioch wrote a letter to the Romans asking them not to intervene on his behalf. He similarly shows knowledge of the various Pauline Epistles, quoting from Romans, I & II Corinthians, Ephesians, Colossians, Acts, and the synoptics. See, for example (I added the verse references):

For if I be truly found a Christian, I may also be called one, and be then deemed faithful, when I shall no longer appear to the world. Nothing visible is eternal. For the things which are seen are temporal, but the things which are not seen are eternal. (2 Cor 4:18) All the pleasures of the world, and all the kingdoms of this earth, shall profit me nothing. It is better for me to die in behalf of Jesus Christ, than to reign over all the ends of the earth. For what shall a man be profited, if he gain the whole world, but lose his own soul? (Mark 8:36, Matt 16:26)

Even in the original epistles themselves, we see an expectation that they will be copied and passed to sister churches ("And when this epistle is read among you, cause that it be read also in the church of the Laodiceans" –Colossians 4:16).

tldr; The 4 gospels and the Pauline epistles were bundled together and widely distributed very early in the church's history, with Acts linking them together. A Christian from one city could write to another with the assumption that they were both familiar with these texts.

You may also get a good response by posting this in /r/AcademicBiblical.