Can migration from MS Access to SQL Server or ... make database faster?
We can't really answer whether you should consider other platforms such as mongodb - we know nothing about the application or its data and whether such a platform would be appropriate. 

I can tell you that SQL Server has a lot more features geared toward performance and tuning than Access will ever have. Access is pretty simplistic and, while it can function as a multi-user database, it is not a true RDBMS and really is meant for the desktop space. 

Whether those features in SQL Server will make your company's specific application any faster, again, shrug, we just don't know.