SQL Server Express to SQL Server Standard upgrade?
Do I need to upgrade to SQL Standard Edition?
  Our website has been very slow and unreliable lately


Depends.

If your sites are still working then that implies that SQL server hasn't stopped accepting new data. I would expect exceptions along the lines of "Could not allocate space because the '&lt;FG_NAME&gt;' filegroup is full" if you were hitting the 10Gb limit already.

The 10Gb is per database not per server, so if your 38Gb is from several databases you may not have hit the limit yet. Also it doesn't count transaction log files so you might have massive log files (because your backup and other maintenance plans are not suitable for your recovery model for instance) that are responsible the majority of the space consumed.


  Is it expensive to upgrade?


This is not the right place for licensing/pricing information - such information changes regularly so any answer would quickly become obsolete and therefore incorrect. For current absolute prices consult your preferred software dealer(s) or the manufacturer directly. For a view on what is expensive and what isn't consult your accountant!


  Would the web hosting company be able to provide us with the license?


Depends. You'll have to ask them that. Some will, some won't. We don't even know what sort of hosting arrangement you have so we wouldn't be able to help even if we were the right people to ask.