In the US, if you are working with biomedical related work, you can apply K99/R00 from NIH regardless of citizenship status, maybe also the human frontier science program. There are also some private research grant/fellowship depending on different subject fields. If you are US citizen, many more federal level postdoc fellowships are open in NIH, NSF, DOE, DOD, etc (e.g. F32).
Hello Yang and thank you for your kind update and yes I am working towards the F32 now and yes I will research the others. Would you know the applicant to award ratio for any of these eg. the F32 applicants per year vs. #s awarded the F32 that year?