That's the thing. It depends on the weather. I haven't bought anything from him when I came across him but he sells ancient cores on rare occassions. Some people said he sold multiple ancient cores and others said they could only buy one.
You can buy them from Teli. You can find him at Hateno but he's also seen at Dual Peaks stable. The merchants seem to seel more rare items when it's raining though. So the chances to buy an ancient core from him might be higher when it's raining.
Another way to collect them is to go to Hyrule...