Instead of spamming the comments with updates, I thought I'd put them here.

The AI seems to struggle with vehicle repairs when it comes to forward repair sites. I first noticed it on a large, modded map but later chalked it up to the map itself causing most of the issues. Later matches I noticed the AI functioning normally, until I played the British again.

In the aforementioned match, it was 4v4 and I was British with the Armoured Battlegroup. It was a long match and our team was winning the tug of war, until my allies ran out of usable armour. A large cluster began to form in two locations, each where I had a Forward Repair Assembly. The AI tanks would cluster there for repairs but would not stop moving. The enemy team eventually pushed quite a bit and I was forced to use five Royal Engineer squads to chase after and repair these tanks that wouldn't stop moving.

The later matches seemed to be fine. The next long match was three hours and was also a modded map, but the AI didn't seem to have a problem with repairing their vehicles. They would either send them back to base for repairs or seek out a nearby engineer squad. So it couldn't have been the map.

The next match I played was the same map, but different team compositions. I was British Armour again, and built a couple Forward Repair Assemblies. I noticed my ally would again begin to cluster damaged tanks nearby and keep moving around, making it impossible for the engineers to repair the vehicles having requirements that they're stationary.

I quit the match and started a new one on a smaller map as the U.S. to try their Forward Repair Station. The U.S. FRS is capable of repairing vehicles that are moving, unlike the British FRA, but with how much the AI moves their vehicles about it doesn't matter too much. Fortunately, if you choose the Mechanized Support Station upgrade, you can convert the FRS into an aura, which will allow allied vehicles to continuously be repaired even while moving. Sometimes they will stop within the aura, other times they won't. In all cases vehicles return to combat when their health hits at least 75%.

• Update: Allied vehicles still struggle to locate the repair aura of the FRS, randomly moving in odd places outside of it. Sometimes they find their way inside but it's still a 50/50 if they stop or continue moving.

I have not tested the German Repair Bunker, but I assume it behaves the same as the converted U.S. Forward Repair Station, both having an aura and being able to repair moving and stationary targets alike.

• Update: I tried the German Repair Bunker. It's an aura, but only repairs while stationary. The U.S. Forward Repair Station's aura is better.

There appear to be no repair issues so long as no Forward Repair Stations/Assemblies are set up. However, I do remember in Company of Heroes 2 where if you began repairing an allied vehicle they would eventually come to a stop to allow you to do it. This isn't the case here, as vehicles will only stop for repairs if the vehicle owner is conducting them.

• This is actually not true. They will stop and allow you to repair the vehicle but will begin moving again once they hit 75% health (or criticals are removed) instead of allowing you to finish repairing them entirely.
  
  Myself and an ally were both repairing a separate tank at the same time. Mine had a critical that required 90% health to be removed. Both the AI vehicle and engineer team returned to combat once the vehicle was repaired to 75% instead of finishing repairs. Mine began to return to combat once the critical was removed at 90%.
  
  If being repaired, shouldn't the team repair it to its entirety? This is why I posted the wrong info, because it felt like they wouldn't let me repair them.

The AI does not appear to repair their Headquarters. I first noticed it during a 4v4 that took a while to win. We finally destroyed one of the AI's bases and I decided to camp outside the HQ territory to ensure the flank was secured. However, no troops came from that direction. I sent a recon team and the entire base was still destroyed about twenty minutes later. This was an official map.

In Company of Heroes 2, if you lost your HQ you got a temporary Call-In to send some engineers to repair your HQ. I am unsure if it's the same in CoH3 but I assume it's similar if not the same.

I paid no mind to it until a later match where the enemy destroyed one of my ally's HQ buildings and did not repair it. I had to send my own engineers to conduct repairs. This was on a modded map, so I figure the map type doesn't matter.