How to get data for STEM/Non-STEM unemployment in USA for 2005 and ahead?
You can calculate this using data from IPUMS CPS or IPUMS USA (ACS data). The ACS has a larger sample size, but the CPS has more frequent data (monthly as opposed to yearly). Both are available from 2005 to the present. The following applies to both datasets. There are different ways to define STEM, but a standard one is the one used by the Bureau of Labor Statistics (link). You can convert the SOC codes used in that list to the occupation codes used in the CPS or ACS by using the crosswalks available from the Census Bureau. These can then be used in conjunction with the variable EMPSTAT to calculate unemployment rates for STEM and non-STEM occupations.
If you need assistance navigating the IPUMS extract system, I suggest checking out these video tutorials. IPUMS also maintains a set of data training exercises that can help with setting up the data and getting the right calculations in your statistical package. Note that you should be using weights when calculating these rates (PERWT for ACS or WTFINL for CPS).