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][1]:

> 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.


  [1]: http://linuxmafia.com/~pam/ACOS_signings.html