Help me solve this problem: giving third party an easy way to create & update records in our database

We are building software for a client that requires a variety of third party website providers to be able to create & update records in our client's database. The gold standard has always been some sort of feed implementation which looks like this:

1. Build a system to consume a feed
2. Ensure each feed has a unique reference
3. Create documentation for third party to build a feed to our specification
4. Get them to provide a URL to their feed (json, xml, or something like that)
5. Enter URL into our system and consume the feed to populate database

However, we've been toying with the idea of instead simply building a REST API, along with the appropriate documentation, which allows the third party to consume and ultimately POST & UPDATE records in our database. There are limitations to this however, compared with the feed approach.

I am looking for some opinions, tips, tricks, tutorials or ideas for this kind of implementation. Have you every built something like this in the past? How did you do it? What would you suggest is the best approach for building this kind of system?