How can I select an HTML tag's content in Vim?

You can use vit to visually select text in a tag. See

:help v_it

Here are the steps:

1. Place the cursor on the tag.
2. Enter Visual mode by pressing v.
3. Select the inner tag block by pressing i+t (or a+t for outer tag block).

at a <tag> </tag> block (with tags)

it inner <tag> </tag> block

See more at "How to jump between matching HTML/XML tags?".