Are there alternatives to Microsoft Access? In my office, we often use MS Access for many applications (eg creating reports, data analysis and data processing) because it is convenient to use (easy data import, fast query building, easy automatization with VBA)

Unfortunately, it starts to be insufficient for me due to the size limitations of the database file. This is one thing. Another thing is that Access is often unable to execute my complex SQL queries (I omit that the SQL used by Access is limited).
Is anyone out there aware of any tools similar to MS Access that can be used with databases over 2GB?

Possibly, I am considering using Access as a front-end, but what about the back-end? Can you recommend something?
Check out Linked Tables to SQL Server if you're running up against complexity requirements for your data.  Basically, as you are expecting, Access would still work as your front-end and SQL Server would work as the back-end.  For complex SQL statements, you can make use of Stored Procedures that you call from Access as well.  If nothing else, this could give you some more time to rewrite the Access application in something a little more robust like .NET (language/platform of your choosing).  Honestly though, if there's no pressing need to change out the Access application, keep it around if it's doing the job.  I support clients that run tons of business functionality off of an Access front end tied to a basic availability group on SQL Server Standard Edition for high availability.  It does the job just fine and it doesn't cost an arm and a leg.