Do PCB fabrication houses bootload/upload firmware?
Yes, this service is available.

It wouldn't be done by the PCB fabricator. They just etch pretty patterns in the copper and laminate up the fiberglass sheets (and drill holes, coat with solder mask, and apply surface finish).

Your assembly shop might have the ability to do it. 

If not, there are other companies who specialize in this work. Normally they'll program the chips (EEPROMS or microcontrollers or CPLDs or whatever) before they're assembled onto the boards.

Depending on the part being programmed, and the volume required, the chip vendor (or a distributor) might also be able to provide the chips pre-programmed.

If your assembly shop doesn't do the process in-house they may be able to sub-contract it to a specialist shop.

In my experience, the process is called "firmware loading" or "EEPROM programming".