It takes most people six months of dedicated effort to put together a K award application. The structure of the grant is such that you have to identify a mentor team and construct a career development plan, which are specific to whatever institution you will be at. Which is why every person I know who has submitted a K, did so in the first 2-5 years after graduating residency, after they got a job as an assistant professor. Logistically, would be nearly impossible to arrange this as a resident, even if it was at your institution, even if you were a MD/PhD.
And the NIH award cycle is such that it will be at least 8 months after you submit before you get your first grant dollar, assuming you get funded on the first try (only ~20% of all K-award applications get funded). Resubmitting would push back that start date another 8 months. Which means that nobody is going into a K right out of residency, even in a four-year program, unless you apply in PGY-2 (unheard of).
K's also require a publication record. It is hard to take a single paper from conception->publication before you graduate residency, let alone the multiple papers needed for a K. In our three year program, 90% of pubs coming out of our resident-initiated projects are accepted and published ~1yr after they graduate. Bottom line: most work done during residency won't be on the CV in time for someone to graduate with a K award.
And are you sure you mean K01? Because most clinicians submit K08s or K23s. I recall the K01s being primarily for PhDs or for people who are coming back to lab after leaving for a few years, not clinicians.
So, if you're "Deb Houry good," you might be able to get a K out of residency. But I think it would be unrealistic for anyone to plan their career that way. A much more common pathway is to do residency followed by protected time in fellowship.