Yes, hospital diploma programs do still exist, but they are far fewer in number than they use to be. The hospitals in my area have made the transition to wanting "BSN preferred or required" for most RN jobs (Texas, near Dallas.) Most of the hospital diploma programs in my area have closed their doors. I would check your State Board of Nursing website. On the website for the BON for Texas, I was able to find a list of accredited RN programs (diploma, associate, baccalaureate.) This list included one hospital diploma program for the entire state that is accredited. Your state may be different though... Could have more options for accredited hospital diploma programs. Best of luck on your decision!
There are still plenty of hospital-based diploma schools around, but, as noted, far fewer than there used to be. Some states no longer have any diploma programs at all, but there are also places where they are still popular (I recently relocated to PA, and PA seems to have as many diploma programs as the whole rest of the country combined ). North Carolina, where I spent most of my career, is down to just two in the entire state (but both of those programs are going strong). It just varies from state to state.
I graduated from a diploma program many years ago. Since then, I've completed a BSN and a Master's and have taught in ADN and BSN programs. I must say I got a much better nursing education in my diploma program than students in any of the ADN or BSN programs with which I've had experience since then have gotten.
My school is part of a hospital. Most of its programs are medical in nature. It is a fully accredited college that offers associate degrees (not nursing-rad tech, respiratory tech, Etc.), bachelors degrees (BSN, biology, liberal arts), and masters programs (MSN, CRNA, PA). When I graduate I'll have a BSN. You still have to take prereqs and apply like any other nursing school. The school has been around since 1909 as a diploma program. Twenty years ago, it became an ASN program, and now a BSN program.
The only diploma program left in the state is part of the hospital I work at (not my school). You have to take prereqs at a local college/university and then apply. It's quite competitive and only accepts 30 students a year and has carried a 100% NCLEX pass rate for the past several years.
In a nearby city, a former diploma program is not hosted by a local community college. They still do all of there clinicals at the hospital that the school originated. Many of my instructors are from that city and went to that school and brag about those graduates "hitting the ground running."
My point of this post is that yes there are hospital -based training available. They may not be exactly what you think of when you say "hospital training." A lot of the diploma programs changed and moved in order to keep up with degree advancements.