I too have an only child, a 10-year-old daughter like yours! First, feel totally fortunate that she can play by herself and entertain herself just fine. These are important skills; no one of any age should just have to be around others or depend on others for entertainment all the time. The ability to entertain herself, without moping around saying "I'm bored," is important.
We all want our kids to have fun on vacation. But I usually find that kids will home in on each other. Hotel pools usually have other kids (though you can't choose ages, or whether the other kids are there in a gang of kids and arent' interested in including your child -- well, that's life, and kids need to learn that). Bringing and sharing pool toys, beach toys, games, etc. is a wonderful thing to do and breaks the ice perfectly.
Also be sure to visit some places that likely will have plenty of kids: Museum exhibits that are kid-friendly and interactive; historic sites that have kids' trails and activities, etc. It doesn't have to be all pools, playgrounds and beaches for kids to find other kids. National parks have kids' programming and terrific "junior ranger" programs that let kids earn patches and badges; many historic sites have scheduled kids' activities; museums have caught on that kids love hands-on areas. Get her there and let the rest just happen.
She'll be fine. I love vacations with just the three of us. It's great to have alone time with your kid, and it's not essential to provide playmates; just try to hit some spots where kids might be present, and encourage her to share those experiences with other kids. She may actually enjoy doing more grown-up things with you at this age, so she may balk a bit if you prompt her to go talk to other kids, yet on the next day, she may be especially outgoing with kids who are strangers. Go with her flow and enjoy.