The full list of trainers capable of serving as merchants is as follows:
Base Game
- All but one of the trainers at the College in Winterhold also sell various magical supplies; typically books related to their school, robes, and soul gems.
- In addition, several of the non-College-affiliated, magic-related trainers are also merchants.
- Conjuration has Falion.
- Destruction has Sybille Stentor and Wuunferth the unloving.
- Illusion should have Atub, but unless you have the Unofficial Skyrim Patch, she is bugged, and not properly added to the factions that would enable this.
- All three Alchemy trainers are also alchemy merchants.
- Similar to Alchemy, all three Smithing trainers are also blacksmithing merchants:
- There are a few assorted NPCs who are both trainers and merchants outside the above.
- Gharol in Dushnikh Yal is a Heavy Armor trainer, and also a blacksmithing merchant.
- Grelka in Riften is a Light Armor trainer, and also a merchant of various items, mostly blacksmithing-related.
- Ma'jhad, of one of the khajiit trade caravans (going between Windhelm and Solitude), is a Lockpicking trainer, and also a fence with the thieves' Guild.
- Revyn Sadri of Windhelm is a Speech Trainer, and also owns a pawn shop.
- Lastly, the following trainers are normally not merchants, but can be married, which will cause them to open a small store.
Dawnguard
Dawnguard adds three new trainers who are also merchants.
- Ronthil, a Speech trainer in Volkihar Keep, who sells fairly standard magic-related items
- Florentius Baenius, a Restoration trainer who can be recruited into the Dawnguard, and sells some assorted ingredients, potions, and a couple of spell tomes
- Sorine Jurard, an Archery trainer with the Dawnguard who mostly sells crossbows, bolts, and the like.
Dragonborn
Dragonborn adds three new trainers who are also merchants.
- Milore Ienth, an Alchemy trainer in Raven Rock who is also an apothecary
- Talvas Fathryon, a Conjuration Trainer in Tel Mithryn who also sells tomes and soul gems.
- Neloth, an Enchanting trainer in Tel Mithryn who also sells staves and soul gems.
Now, this all may be a very large list, but it's unfortunately not a very promising one. The best choices for selling goods to are generally Fences, as they have the most money (depending on your Thieves Guild side job completion), and also buy anything without need for the Merchant Perk. But the only Fence who is a Trainer is Ma'jhad, who is part of a Khajiit caravan; if you find yourself needing to wait around for his gold to replenish at any point, he will likely be in a different city when you're done.
Your next best bet is probably going to be a blacksmith, as several of them have a thousand gold (including all three who are also trainers), and they buy many of the items you're looting off enemies, even without the Merchant Perk.