How can I add a second foreign employer (no W-2) to my income tax form?

@ezraswife , from your post I am not  sure :

(a) you are a US person

(b) that your tax home is in USA i.e. you live and work in the US

(c) if the foreign employer  has a IUS base 

Generally, if you a US person ( citizen/ GreenCard ) working  in the USA for a foreign employer ( with NO  local; office/ presence ), you show your income  on Schedule-C  --- sole proprietor/  contractor/ consultant  self-employed -- you are liable for Federal / State and FICA-SECA taxes.

If you are abroad ( US person as defined above ) again you enter  your incomes  just as if you are a  self-employed and utilizing the schedule-C and schedule-SE  ( as above ).  Whether you have multiple employers or you move from country to country ( and  with different  employers  or otherwise ), you still follow the same procedure.

Granted , my response is kind of general but  unless you provide more specifics ( your immigration status, whether your tax home is US or foreign etc. ) I cannot be more specific.   If your case is  one that is not of general  interest , then you are quite welcome to send me details  of your situation  and/or questions via PM  ( please no personally identifiable info ).

pk