There are many opportunities, but people generally don't take advantage. Most "basic science" work that radiologists do is really translational rather than bench/animal work in The traditional basic science sense.
Basic science in radiology is mainly physics (looking at new mri pulse sequences, new PET tracers, etc). There are abundant opportunities but it is extremely different than what you currently consider to be "basic science research"
You can find radiologists doing all kinds of things, but it is more difficult: for an internist doing basic science research, they can pay their own salary from grants. For a radiologist, the salaries often exceed the cap placed on total income allowable from grants, therefore you cannot pay for yourself from research. Since the clinical time generates more RVUs than the money you bring in from grants, many departments are reticent to give you the kind of protected research time it takes to run a successful basic science program.
All this means is that you have to get a job at a place that does research for research' sake, acknowledging that it is a money pit. Most of the top rated academic programs support such research. If you look at the faculty pages for Wash U, MGH, Michigan, Penn, Hopkins, Duke, etc. (and I assume the top West Coast programs, but I'm less familiar with those) you'll find plenty of radiology faculty doing basic research, ranging from animal studies to cellular work to develop molecular imaging targets to collaborations with cardiologists on mechanisms of atherosclerosis to imaging of metabolism in different tissues. Wisconsin and Yale also have diverse research facilities, and UNC is working on a big new imaging research center.
The bad news is you have to be very competitive to get to most of these places; the good news is that only about 10% of radiologists seek out academic careers, so there are jobs for you if you're good.
(should qualify a little - this is not to say there is NO basic research except at these schools, only that it's hard to find a variety of basic research available elsewhere. Also, know that if you choose a research heavy career, you'll earn less than your private practice counterparts - this gap tends to be wider in radiology than other specialties).
You can do whatever you'd like. But you have to compromise for it and then you have to see whether it is worth it or not.
If you want to do basic science research you need to have the tools required for it. Medical school is not the place to do it. So probably you need to have a PHD.
I have seen people with anatomy or physics Phd in radiology. You should work in an academic center with secured research time. Doing a basic research needs at least 3 days time dedication. You can do it if you work hard.
The truth is, generally in medicine, people do not last long in basic research, though it is doable. Excluding some outliers, even if you are a family doctor, working 4 days a week in an outpatient clinic from 9-5 pays more than a research job 5 days a week with much much less hassle. As a result in the long run people with MD eventually become more clinical with clinical research. In addition, clinical research appeals more to MDs. After doing clinical medicine for 4+6 years/med school+residency you develop a very clinical and applied mindset.
There are always some exceptions. The best example is Dr. Elias Zerhouni, a radiologist who was the director of NIH from 2002 - 2008. He did some great basic CT and MRI research in 80s and 90s at Hopkins.