As long as you are spending at least 50% of your time doing research, then it's definitely worth a shot!
The odds of success are posted on the LRP website - you have to click on each individual program to see the funding history. For example, for the clinical research LRP the numbers are here:
http://www.lrp.nih.gov/about_the_programs/clinical.aspx
The data are different for pediatric research, health disparities, etc.
Although you could technically apply as a resident, I think it would be rare for a resident to have at least 50% of their time protected for research for 2 years. As a result, you're probably not going to be eligible until you reach fellowship - and even then, it would have to be arranged that you would have 2 years of at least 50% time devoted to research.
With all of that said, the application is very heavily weighted on the strength of the applicant, your potential for a research career, and the environment/mentorship that you have available to you. So taking a little extra time to get a few extra publications under your belt will certainly help. Conference presentations are not included on the NIH biosketch, so publications will be very important here. Also do not underestimate the importance of strong letters of support from your mentors.
Hope that helps! It really is a great program.
