How do I enter an emoji into a string in Vim?

You do not need to install a plugin for this. All you need to do is enter the unicode value of the emoji. You can do that in insert mode with <C-v>. From :h i_ctrl-v_digit:

                        *i_CTRL-V_digit*
With CTRL-V the decimal, octal or hexadecimal value of a character can be
entered directly.  This way you can enter any character, except a line break
(<NL>, value 10).  There are five ways to enter the character value:

first char  mode         max nr of chars   max value ~
(none)      decimal        3        255
o or O      octal          3        377      (255)
x or X      hexadecimal    2        ff       (255)
u           hexadecimal    4        ffff     (65535)
U           hexadecimal    8        7fffffff (2147483647)

For example, if you want to enter this smiley-face, which has a unicode value of U+1F60A, you could simply type:

<C-v>U1F60A<esc>

or if you don’t want to hit <esc>,

<C-v>U0001F60A

Just so you know, there’s a good chance that it will not render properly in vim, depending on your font. If you are using gvim, you can change :se guifont=*, or in regular vim, changing your consoles font to make it render (assuming you pick a font that can render this particular emoji)