No, he's not her father. The story seems to hint at that a bit, but the timeline doesn't line up.
In particular, in The Shadow Rising, Moiraine is showing off how much she knows about Thom, and she claims that he was:
Courtbard at the Palace of Andor in Caemlyn. Morgase's lover for a time, after Taringail died.
If we take her at her word, then Elayne was born before Thom became Morgase's lover, so he can't be her father. It's possible that Morgase was just covering up Elayne's parentage to protect her, but that kind of thing would be very hard to keep quiet about in a royal court -- there would at least be rumors. And if there were rumors and/or any doubt, and Moiraine knew it, she wouldn't be able to make such a statement without hedging her words a bit.
I believe there's at least one POV chapter from Morgase where she also thinks about Taringails "two children", and obviously she would know.
Out of universe; Robert Jordan explicitly said Thom was not Elayne's father at one of his book signings:
QUESTION
Is Thom Elayne and Gawyn's real father?
ROBERT JORDAN
No. Absolutely no. No question about it. No. No. No. Thom is exactly who he says he is.