Can I Use SQL Express to Migrate a customer's system on my local machine and expect it to work when we port it over to their SQL Server?

There's a very good chance that everything on SQL Server Express will function the same in SQL Server Standard or Enterprise. But I still wouldn't develop on Express if I were you.

SQL Server Developer Edition costs just a few dollars more (typically < $50), gives you a much better set of tools to work with and has all the same features as Enterprise. With Developer Edition you'll be able to take advantage of and test features that aren't available using Express.