Full disclosure, I work for Peptriva (research-use-only compounds), so factor that in. Can't help with anything use-related but COA stuff is basically my whole job, so.
Honestly you kind of answered your own question. If you have to go hunting or email a vendor to get a COA, that tells you most of what you need to know. Vendors doing it right just put it on the product page, per batch.
If they do send you one, look for the lot number first. No lot number means it's a generic doc they could be reusing across everything. Then check it actually names the methods, LC-MS for identity and HPLC for the purity number. Endotoxin and microbial should show up as their own line items too, the purity % doesn't cover those. One thing people miss: if every product on a site comes back exactly 99.9%, that's worse than seeing a bit of spread. Real results vary.
And after the whole Paradigm thing, worth remembering a vendor's own COA can just be made up. The only thing that actually settles it is independent testing, either an ISO 17025 lab or customer-submitted
Janoshik results. Same standard applies to us, for what it's worth.