As I understand it, most of what we know about Norse mythology is based on texts written in sagas by Christians in the late Middle Ages. Do we have any contemporary accounts of Norse mythology from the period in which it was still believed?
Aside from some short inscriptions on stone, no.
The received texts of the sagas generally all date to after about 1000 A.D. and were written or copied at times when Christianization had taken hold.
That said, however, it is important to remember that it is likely that the received texts may, in many cases be close copies of manuscripts written during pagan times.
Also, do not assume that Christianization was universal. In many cases there were scribes and others who were only nominally Christian or sympathized with old customs. Just as a single example of this, Yule continued to be celebrated in Scotland for centuries and centuries after Christianization.
First of all, it should be noted that Norse mythology was only the longest-lasting offshoot of a much wider spread Germanic mythology. If we restrict ourselves to Norse contemporary sources, then there are only short inscriptions and pictures.
For the Germanic mythology, there is one contemporary non-Christian source that must be mentioned: Tacitus. In his Germania (98 CE), he writes a bit about Germanic deities. A lot of it is Interpretatio Romana (Tacitus used the term himself), meaning that the dieties are identified by what was hopefully decently close Roman counterparts. However, there is one long passage which is not related to us in terms of Roman gods. This is the description of the goddess Nerthus, who had a statue too holy to touch, which was driven around to different places and afterwards washed by slaves who were then killed. Nerthus is most likely related somehow to the (male) god Njord, which are mentioned in the Norse sources.
To return to the strictly nordic, most of the knowledge of the mythology has reached us through the poetry of the Older Edda, or through Snorri Sturlason, who wrote an instruction so people whould better understand Scaldic poetry, which contains almost all of the mythology you would find in popular books on the subject. The Norse sagas usually don't deal much with the Norse gods. The sagas are either overtly Christian or don't mention religion very much at all; the Norse gods are seen as demons when they take active part. They do, however, sometimes give insight in how the cult worked.