A coworker and I are putting together a simple inventory of companies and products in a certain field, so we're looking at putting the info into a simple database with a couple tables (companies & products, essentially). This is opposed to putting it into an Excel sheet or just making a list in a word processor.

But we both need to be able to contribute to this database separately and be able to access the complete database.

Note that running a mysql server or something is out because we're both using laptops we take home after work, and we don't have access to a database server at work.

Is there a way to do this with OpenOffice Base, or with some kind of web-based service?

You can connect mySQL server with OpenOffice Base so you will implement the database from there.
You can even build a database with OpenOffice Base as you would do with MS Access.
But I can't understand how you would like to access the complete database at the same time both of you. You need a server running for this... Maybe you should put on a mySQL server at work.

Yeah, I'm seeing that it's going to take some kind of server to make this work. Right now we're trying Google Spreadsheet since it's such a simple dataset, but if that's too restrictive we'll have to work on a server somewhere.