"When I was reading on the subject I found a source that said the orbit was estimated at 80,000 years. One said it was unknown. One said it was uncertain if Proxima Centauri was even a part of the Alpha Centauri solar system and suggested that it could even leave orbit after a few million years."
The most recent analysis I am aware of (2006) suggests that Proxima Centauri is part of the Alpha Centauri A/B system and that 'Proxima [is] currently located near the apastron position of its orbit.'
If this analysis is correct, then Proxima is at or near its most distant from A/B. The entire paper can be found here:
There is still uncertainty as to whether Proxima is in orbit about Alpha Centauri A/B, but if it is, then its orbital period is quite long.
I wouldn't worry too much about the period; Avatar has a gas giant in the A/B system and latest findings suggest that is very unlikely; rocky planets could form in the A/B systems, but their orbits may only be stable for around 250 million years, and then only if in orbits of less than 2 AU from the parent star.