Are there any vintage SxS trap guns with a reputation as being safe, fun and reliable for a trap shooter on a budget?
Your question brings several thoughts to mind.
1. First, you do not need a "trap model" to be successful shooting trap. Some of the best trapshooters in the world, Dan Bonillas for one, prefer flat shooting guns.
2. As others have replied, a vintage gun, in a "trap model" will cost extra bucks over the same gun, in a field model.
3. A very old trap model will have the same safety concerns as a very old field model of the same gun.
4. The Daly mentioned is very likely an O/U, seller didn't specify.
5. A BSS would be an excellent choice. Strong, modern ammo compatible and reasonable. 12 ga BSSs usually are in the $1000 range, or less. The Bernardelli Hammer Gun listed in the for sale section would be another good choice. If you want to be modern ammo safe, you'll need to stay fairly new, say 50-60 years or so.