I wouldn't worry about your frame being strong enough. I've never heard of anyone breaking a frame due to load. People have ridden WalMart bikes across the entire USA.
There's a small chance your wheels may not be up to the task, but as long as they're not some super low spoke count wheels, they're probably ok too. You might have a good bike shop stress-relieve the spokes and tension them properly for insurance. That's a lot cheaper than buying new wheels.
You can mount racks on bikes without eyelets using P-clamps. See
this article.
You may find the price of inexpensive racks and panniers almost equal to the BOB Yak trailer. I prefer racks and panniers, but lots of people tour happily with a trailer.
I suggest making your first tour without investing too much in specialized equipment. You'll get hooked on touring for sure and your experience will help guide you to eventually putting together the purpose-built touring bike that meets your long-term needs and wants. For now, just ride and enjoy.
Edited to add:
There are a number of rack mounting solutions for bikes without eyelets at
this site.