Yes, you can use WayBack machine to recover the GIF of deleted blogs but it is not a reliable way. If you want to know then follow the steps:

Visit the official site of Internet Archive

Then enter the exact URL of your deleted blog

<!-- notionvc: 9be82c39-8bd6-48ce-8816-b22e5ca96b42 -->

Now the WayBack machine shows the previous snapshots of your page and you can find out the GIF by navigating to the different time period when the blog was active.

To know more such methods of How to See GIF Packs for Deleted Blogs Tumblr then read the blog.