If you don't have a Tumblr account, you can't use the lookup feature available to registered users. Since there's no way to directly search for friends on Tumblr if you aren't signed up for the site, you'll have to rely on some detective work to track down a friend's Tumblr account. With enough perseverance, you may be able to find it.
If your friend wants to share her Tumblr, she can give you a direct link to her blog. You'll be able to see all the entries she's made as long as her blog isn't set to private. Bookmark the blog so that you can return to it in the future without having to ask for the URL again.
Even though you can't search Tumblr posts without an account, you can still search for specific items on Tumblr. For example, you can look for a friend's name by typing in the name followed by "site:tumblr.com" in a search engine. It will display Tumblr pages that include the name you used as a search term. Search for user names, nicknames and other key words that might help you find your friend's Tumblr blog.
Your friend may connect her Tumblr account to her other social media profiles. Scan her Facebook, Twitter and Google Plus accounts to see whether she lists the URL for her Tumblr. She may also share posts on social media sites that link back to her Tumblr, even if she doesn't have the URL shared on her personal information. If you see any links to Tumblr, click on them and view the linked account to see whether it belongs to your friend.
If your friend has a rare interest that isn't shared by many other Tumblr users, you may be able to search for that term on Tumblr and find posts made by your friend. As you scan through posts, look for familiar pictures, posts that sound like they were written by your friend or usernames that you recognize. Some Tumblr users upload a real photograph of themselves as an avatar, which can also help you recognize a friend's account. Make sure to specify that Tumblr is the site you want to search on when searching via a search engine.