It sounds like you're discovering the same thing I discovered, except I discovered it a little too late-VERY few positions in a veterinary situation are going to want to/be able to pay you.
When I was your age (I make it sound like that was forever ago, haha really it was 2 years ago) I spent a lot of time trying to find a veterinary assistant position that would pay me. I applied to 4 vet clinics and came up with nothing (though to be fair, only 1 was actually "hiring"). None of them hired me but I couldn't work for free because I had horses to feed. I needed a paying job. So I did the best I could taking positions cleaning stalls, doing kennel work, etc. It wasn't veterinary experience but it was the best I could do.
Unfortunately, going to school full time, working part time, and taking care of my animals the rest of the time left very little time to do any slave labor/shadowing at vet clinics. So I didn't do a lot. Until last year, when I realized, oh SHOOT, I need to get going on my vet hours!
So now, here I am, about ready to submit my application and I have less than 500 hours of directly supervised vet experience. Mind you I have thousands upon thousands of hours of animal experience, but my vet experience is obviously lacking so I'm really going to have to make up for that during my interview, should I get one. If I could go back knowing what I know now, I would totally not have wasted my time looking for all those vet positions and just taken the stall cleaning/kennel jobs a lot sooner, and then saved a day or two a week to just shadow any vet that was willing to let me hang around.
That being said, I DID get about 100 hours of vet experience working at the dog kennel, because I got to help the vet with exams on the dogs that were staying with us. So as was previously mentioned, you might be able to get your foot in the door working at the kennel, so it definitely can't hurt.
My advice would be the following: take the kennel job. Do the best you can, think of every possible interaction with animals and fellow employees as another step towards vet school. Continue to do well in your classes, and if you have one or two days a week with a few free hours, find a clinic that will let you shadow there. Because honestly, you're going to have a pretty hard time finding a clinic to pay you with little experience, but every experience is valuable to you, whether you're paid with money or not, because every hour spent with a vet is another hour you can put on your VMCAS. Most vets are more than willing to let you hang out.
Good luck!