Speculating: Yes, but it's probably a bit harder to make a backdoor in an FPGA that can anticipate and manipulate a custom design. The backdoor would have to be smart enough to know it's running a CPU, although if it could just detect that it's running multiple Ethernet ports, it could still act as a silent forwarder for evil packets or a data mangler. It could conceivably also try to send the FPGA design to the backdoor's owner for them to tell the backdoor what to do next.
Yes, the fab could put in a backdoor. That's why you need to build your own. But, what if the machinery you purchase has a backdoor? Now you need to build your own fab machinery! It's backdoors all the way down. There always has to be some level of trust.
Doesn't really matter, though. If you have to build your tools up from that level, it's going to cost so much that only governments and huge corporations can do it--and those are exactly the people installing the backdoors in the first place, so we're back to square one.
