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Privately, she almost certainly does - most of the Royal family are avid hunters and shooters. The Queen no longer shoots herself, but she did for many years.

But on that issue, as on others, unless the government acts outside the constitution, she cannot act. And while a lot of the British constitution is unwritten, the fact that gun laws are in the hands of Parliament rather than the Monarch is actually one of the relatively few bits that is written down:

And thereupon the said Lords Spiritual and Temporal and Commons, pursuant to their respective letters and elections, being now assembled in a full and free representative of this nation, taking into their most serious consideration the best means for attaining the ends aforesaid, do in the first place (as their ancestors in like case have usually done) for the vindicating and asserting their ancient rights and liberties declare ...

.....

That the pretended power of dispensing with laws or the execution of laws by regal authority, as it hath been assumed and exercised of late, is illegal;

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That the subjects which are Protestants may have arms for their defence suitable to their conditions and as allowed by law;

Britain has a constitution. The Queen follows it to the letter, always has, and always will. When that constitution gives a power to Parliament, that's the end of the matter.

Personally, I think constitutions should be respected in that way to that extent.