You can surround the word currently under the cursor in quotes with the following normal mode commands:
iw with any other Vim motion/text object to surround other things with quotes*. Or
"" with any other pair of characters to surround the object with different things.
If you want to surround the object with a longer piece of text, such as an HTML
<p> tag, you can use Ctrl-R instead of the
P put command:
:help i_CTRL-R for more details.
This latter command performs the change as a single edit, and thus is repeatable with the
. command, but as discussed in the comments, when you repeat the command, it will enter the text from the original change command. However, as @user938271 explains, you can workaround this by using Ctrl-RCtrl-O or Ctrl-RCtrl-P instead of a plain Ctrl-R when inserting the contents of the register, making it easy to apply the edit quickly in several different places.
* Or use
c from visual mode to surround text that is hard to describe with a single motion.