Shortcut to select next item in a list
If you look "closer" at the fonts/names in the dropdown list you'll find they are being handled by server:
"patorjk.com/software/taag/#p=display&amp;f=Rectangles&amp;t=Hello\ world!"   

The option group's name (html) is "optgroup label="Featured FIGlet Fonts", one font e.g.:
&lt;option value="Rectangles.flf"&gt;Rectangles&lt;/option&gt;

Therefore I assume they are sent by/to script
&lt;script type="text/javascript" src="./js/figlet.js"&gt;&lt;/script&gt;    

It is certainly possible to delve deep into JavaScript and trace whatever commands it takes to solve this.
Actually I "studied" this page, tried AppleScript with/and Karabiner, but the list's behaviour is so "nervous** " that in the end I settled for a pure Automator/AppleScript solution:

tell application "System Events" to tell process "Safari"
    if name of front window of application "Safari" is "Text to ASCII Art Generator (TAAG)" then
        key code 125 --down… 
        delay 0.1
        key code 125
        key code 36
    end if
end tell


Actually this "just-so-working" code presumes you have selected the list, then press the shortcut shortly, let go, press shortly, let go, ...
(** AS/Karabiner, via osascript, worked nicely &amp; quickly – but only circled through fonts with same 1st letters … very irritating, possibly due to some settings in the js-script.)

Now if this particular page is really important to you, try this: open Automator, goto    

File>New>Service / left window: Execute AppleScript / top lists L/R: No input / Safari (any browser).
Paste above code into window. Safe as "Fonts-down" (for "going up" replace "125" with "126").    

Final step: open Sys-Press Keyboard Shortcuts Services (left) scroll-down to "Fonts-down", check and input your chosen shortcut (e.g.: Opt-D). It should now appear as (menu) Safari>Services>Fonts-down.
(Depending on your general settings you may have to ONCE "allow" this script in Sys-Prefs.)

Okay.
– This script will only work in the specified window (important)
– If you press too long (try it) fonts may be skipped
– Sadly you'll have to choose a multi-key combo in Sys-Prefs keyboard, like Opt-D (Opt-U for "up"?)
&nbsp; &nbsp;(If you use Karabiner you may define a 1-key shortcut to call "Opt-D".)
– This page is difficult to handle as its displayed content is created server-side.