If you simply prefer website.com/blog over blog.website.com, you could set up the former url with a redirect to blog.website.com. That has the added benefit of allowing people to access the site two different ways.
However, if the problem is that you can't change your DNS settings, you could use an iframe. That probably isn't a very good idea though, because it messes with the browser's back and forward buttons and doesn't display the updated url when you navigate within the frame.
Another non-DNS solution, if you have some kind of scripting available, is to use Tumblr's API to recreate your blog in your own page.
Your options are really dependent on your situation, which you haven't told us much about.