Not by the rules, but the DM can allow it
Rules as written, Githzerai characters cannot be psionicists, even if they have a natural bend towards psionics. The Planescape Campaign Setting states (p. 12):
Character classes are limited to fighter (9th-level maximum), fighter/wizard (9th/12th levels respectively), wizard (12th level), or thief (15thlevel maximum). The githzerai revere an ancient wizard-king as their god, and they have no priests
How serious the book is about these limititations can be questioned. On p. 90 we encounter Yangol, an NPC character that is a Githzerai (Mage/Thief)-(W2/T3), something not possible according to the strictures above.
The Planeswalkers Handbook further advises (p. 90):
Prime characters are naturally allowed the full range of class options found on their prime-material worlds. Planar characters are usually limited to the classes found in the Player's Handbook, unless a player presents a compelling argument to the DM to allow otherwise.
and Gith, although decended from humans in the eons of time, count as planar, with their home plane in Limbo. So, they would not be able to become Psionicists based on this, either. This takes us to the "compelling argument" for the DM.
Allowing it anyways
Githyanki and Githzerai are really the same orginal race, Gith, they just train their mental capacities in different ways and have different philosophical outlooks. p. 124 of The Complete Psionics Handbook lists Gith as a monster race option that has access to Psionic powers
I think this natural penchant for psionics is a good argument why, with Gith as a player race, they should be among those races that can become psionicists.
Further more there is this comment, from the Planeswalker's Handbook (p.91)
OTHER CLASSES. Psionicists are basically wizards, as far as
most planewalkers are concerned - although their highly
trained minds are certainly worthy of respect in a place
where the power of minds and beliefs can change the multiverse
Now, this is of course from an in-character perspective and more tongue-in-cheek, they are not really wizards. But, as Githzerai can be Wizards, and from the viewpoint of most in the multiverse that a Psionicist is really just another flavor of Wizard, then the DM could grant that a Psionicist should be fine, too.