How to select a whole HTML block, including the tags in VIM?

vat, as in Heptite's answer is the way to go.

Note that, depending on where the cursor is, you may need to type at a few more times until the whole <div> is selected.

Supposing the cursor is in a <td> you'd need vatatatat to select the whole <div>:

<div>      ^  at
  <table>  |  at
    <tr>   |  at
      <td> | vat

Another option would be to search backward for di(v) or he(lp) and select the whole tag with vat:

?di<CR>vat

Place your cursor within the opening or the closing tag and type "vat".

See:

:help text-objects
:help v_at