I don't know of any residencies that do that.
However, I do know of a couple of post-doctoral fellowships that accept MDs that pay for MPH degrees. A postdoctoral fellowship is usually 2 years of intensive research and training; you work with a mentor who advises you in research and author papers with them, and you usually do professional development and training, sometimes formal classes. I am a postdoc now and looked at several, and many of them offered (and required) their MDs and non-public health PhDs to earn the MPH during their postdoctoral fellowship there. If you want to transition to a research career, a postdoc is probably a good idea anyway. You can do one before or after your residency - I've seen postdocs that take both.
Here are some examples:
UCLA's health services research:
http://hpm.ph.ucla.edu/academics/programs/ucla-rand-post-doctoral-program (the MPH is in health policy and management, though, not biostatistics)
University of Pittsburgh/RAND's health policy postdoc:
http://www.rand.org/about/edu_op/fellowships/ruhpi.html
UNC-Chapel Hill Sheps Center:
http://www.shepscenter.unc.edu/fellowships/nrsa-fellowships/nrsa-postdoctoral/
Northwestern's health services research postdoc:
http://www.feinberg.northwestern.edu/sites/cehs/fellowship/overview.html (not sure if they offer an MPH, but you can do a master's offered by the graduate school).
UCSF's AIDS postdoc:
http://caps.ucsf.edu/training/taps
University of Washington's in health services research:
http://depts.washington.edu/ahrqhsrt/
In addition, the University of Washington's Institute for Health Metrics and Evaluation has a postdoctoral fellowship (
http://www.healthdata.org/post-graduate-fellowship/application-process). You need to already have a relatively strong quantitative background for this. The National Center for Health Statistics has one too.
I know Columbia has a few, I just don't know the exact names. I knew a couple of postdocs there doing MPHs when I was a graduate student there. Look especially in epidemiology, which offers a lot of the same rigorous mathematical training in a more applied sense.
Lastly, a lot of postdocs might not formally offer a master's degree but will work with you flexibly for you to either take the courses that you really need to get the training in biostatistics or to allow you to earn a full degree.
