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The technology does exist it's called radar, and there is already a mission on the books (the Jupiter Icy Moons Explorer, or JUICE) which will take just such an instrument to Jupiter and its moons in the early 2020s.

P.S. There's also Europa Clipper, but the funding is less certain: https://en.wikipedia.org/wiki/Europa_Clipper

Radar has also been used to map underground ice deposits on Mars, see eg. [1][2].

[1] https://en.wikipedia.org/wiki/SHARAD

[2] https://en.wikipedia.org/wiki/MARSIS

I don't remember if the moon was the target, but I recall there was a probe in the last 30 years or so that was deliberately crashed into something just to study the debris that was kicked up.

You might be referring to Deep Impact [0], which wasn't even quite that long ago.

[0]: https://en.wikipedia.org/wiki/Deep_Impact_(spacecraft)

or LCROSS: https://en.wikipedia.org/wiki/LCROSS

Going back further, the original impactor was Ranger, I think. Video just prior to impact:

https://youtu.be/dE8pe2EYlZA

I don’t know about bombs per se but plenty of stuff has been deliberately crashed into the moon to study its seismic structure etc.