No I would not use Vit C after doing your TCA. You actually shouldn't be using anything until after your peel is completely done peeling. Just wash, if you need a little moisturizer, neosporin or emu oil, etc you are ok using that. But really, don't use anything else until it is completely finished peeling, and the pinkness has gone away.
I just stuck with my oils and antibiotic ointment after my peels and did just fine. I have a mixture of jojoba, emu, sweet almond, vit e and HA (tiny amount w/water added to the oils) that I use after my peels.