Vom is actually a contraction in German, shortened from von dem. Depending on the gender of the word that follows, it might be appropriate to use vom, but it might also be grammatically incorrect. If the next word is feminine or plural, you would want to use von der or von den instead. ...Actually, depending on exactly what word you're using, you might use a different preposition than von to begin with... (Like how in English we used in/at/on depending on what exactly we're trying to say and what words we're using in the sentence.)
All of that being said, you can name your dog whatever you'd like! You don't have to follow German grammatical rules; you don't have to follow any grammatical rules at all! If you like the sound of something, even if it's not grammatically correct, you can go ahead and use it (as long as it's not on a prohibited list or a registered kennel name of someone else). You also don't have to have a super long name! When I registered my girl, I ended up being able to just straight up register the name I'd been calling her, because it wasn't taken -- no need for a "from" or an "out of" tacked on to the end of it. Just go with what you like best!