We have an established apricot tree and greenhouse in the Bay
Area. So is it possible to take the exiting seeds and grow more
trees with apricot fruits?

The taste (and fruit size, etc) of seed-grown apricots won't
necessarily match the parents.

(well, that's the general answer for most plants which are customarily
propagated by grafts and other vegetative means, I don't really know
about apricots in particular).

You can always try it.

As for getting the seeds to sprout and grow (that is, whether they
need chilling or other treatment), that too tends to vary from plant
to plant (and I don't know about apricots).

Buying a small tree is probably a safer course, but it all depends on
your preference.