ILM was doing GHOSTBUSTERS 2 & INDY 3, plus the pseudopod and the at-time-of-release-unseen tidal wave for THE ABYSS, along with bits for several other films plus a ridefilm.
But they still could have done TREK, and I think it was more about the dollars than the availability (which was the same issue on TUC till the studio decided saving a mil on VFX was not going to be worthwhile, given what had happened on TFF -- and based on the figures HARVEY posted awhile back indicating TFF actually cost more FX-wise than the ILM shows.)
How could they have done
Trek V with that workload? They only had so many motion control rigs and the optical printers were always a bottleneck.
A lot of it has to do with the balancing act ILM has always done, sharing some resources while directing other unused ones towards the project at hand. This was more prevalent in the 90s when you had traditional and CG, but all you need to do is look at TWOK, where they'd ship the vista camera down to L.A. for a half-day to shoot the genesis bubble cave, then send it back up to ILM for whatever they needed it for on POLTERGEIST or ET.
Also, TFF was DESIGNED for most of the work to be done in-camera, even if at the last minute Par got cold feet on doing the God stuff that way (have never figured out why, unless they thought it'd put them back behind schedule again after Shatner had gotten them caught up.)
ILM had already gotten a slight taste of the non-comp in-camera route on TVH with mixed results (whales were awesome, but the BoP under the bridge not so great, with the one big shot of that having to be redone as a not so great optical.) The other part of all this is that ILM KNEW HOW TO DO MOST OF THE WORK ALREADY - they had largely managed to solve issues of shooting the E (even if it didn't look as good as Trumbull's work) and comping it, which clearly was not something the TFF VFX folk had a clue about.
ILM had more optical printers than anybody I think (by way of comparison, the company Ferren used for opticals, The Optical House, had exactly ONE, and because they had to wait forever for Ferren's model photography place to actually get their custom mo-con camera, they couldn't even start shooting those elements till February, I think, leaving no time for learning curve considering the movie was coming out in June.)
The ILM 4-head one Edlund had built for EMPIRE got broken up into two separate printers a few years later, so they usually didn't have a huge backup. The ABYSS work was mostly CG, so that wouldn't have been a major factor on TFF, although they might have used CG for some of the barrier and God planet stuff, but they wouldn't have NEEDED to do so.
There's a bit more to the ILM issue here (which is more about the money and inclination than the resources), as the story has been that Paramount fully expected to be able to reuse a bunch of stock elements and shots from previous films, but most of those couldn't be accessed (except I guess the two cuts that ILM DID do for the film.)
I assume the elements were AT the vfx vendor, so that suggests somebody didn't look too hard, since at least one shot in TUC was an unused leftover from TVH (the rather crappy 'zooming into camera at warp' shot that turns up around the time Kirk & Spock talk about being obsolete is an exact match for the shot described in a mag at the time , probably AmCin, as one of two warp shots done for TVH -- the other being the one that is IN that movie.) If they were all at paramount and couldn't be located for TFF, that sounds like some serious self-sabotage.