Are you talking about during training or after you've completed an MD/PhD?
During your PhD training (ie prior to medical school graduation), some MD/PhD students will apply for F30s grant which a training grants just for MD/PhD students. These grants are awarded to the individual applying (well really awarded to the university, but that is semantics).
After medical school but before becoming an attending, MD and MD/PhD trainees can apply for institutional T32s, K12s or KL2s. These are awarded to an institution and the person applies to the institution (ie university) and not to the NIH directly.
After becoming an attending, MD and MD/PhD apply for either K08 or K23s at the junior faculty level. This provides protected time and some salary support. There are also K01s, which are directed toward PhDs. I don't know if an MD/PhD can apply for a K01, I haven't personally see it (and they are rare awards anyway)
Afterward a K award, attendings will typically apply for R01s (both MDs, PhDs and MD/PhDs).
Here is a full list,
http://grants.nih.gov/grants/funding/ac_search_results.htm and
http://grants.nih.gov/grants/funding/funding_program.htm
In general, K/T/F awards are training awards, R/P/U awards are for faculty investigators/multiple PI/centers
