I am making a warlock and am faced with one of the most overwhelming parts of character creation: languages. I have searched everywhere, but I can’t find a list with every option. What are all the languages? Who speaks them?

The list of languages is on page 123 of the PHB (Player's Handbook).

Languages

It isn't the complete list. However, it is a list of the most common ones.

Quote from Lyxen >>

Note that this is the list of languages for the FR and it's not a complete list either. But it's a good basis for PC languages.

 Actually this is a list of languages common to all settings and excludes any setting specific languages including Faerun specific languages. For example, the list doesn't include languages such as Alzhedo, Illusken or Thayan.

Quote from DndExpert3 >>

I found all [Redacted] list from here. Thank you so much. 

That list is incomplete and misleading (due to the elemental languages).  Here's a more accurate list:

1. As linked earlier in this thread, here is a partial list of languages.  There are 3 languages we need to discuss in detail here:
   1. This is the only reference in the game to Thieves' Cant as a language, but no other rule stops the Cant from being a language.  Thieves' Cant doesn't generally act as a distinct language - instead, it lets you engage in steganography.  Still, no language list is complete without it, because the PHB defines it as a language.
   2. Druidic isn't in the PHB languages list, but is explicitly discussed where the Cant is discussed, which is good, because the Druid class definition defines it as a language.
   3. Primordial has 4 dialects: Aquan, Auran, Ignan, and Terran.  Anyone who knows Primordial can generally communicate with anyone who knows one of the dialects, but e.g. someone who only knows Ignan can't communicate with someone who only knows Terran.
   4. That gets us 17 languages, plus the Cant and Primordial's 4 dialects.
2. For anyone keeping count, that means we have a rogue language in the list you linked, since the one you linked has 22 and so does our list so far, but I included the Cant and your linked list did not.  The "extra" language is Gnoll.  There are a wide array of racial languages listed in various race descriptions spread throughout the game - Gnoll is one of them.  This can get into weird weeds because NPC statblocks are not reliably consistent - for example, a Mind Flayer speaks Deep Speech, but writes things in Qualith, which is like braille but harder - non-Illithids can't read it without magic even if they are fluent in Deep Speech.  Worse, that information isn't in the Mind Flayer's languages statblock - and you know what is?  Telepathy, which has a context-dependent definition - various ways you can find to become telepathic will have different rules for communicating.  Here's a partial list of racial languages and pseudo-languages, but I'm basing it on statblocks, so I can't guarantee I'm covering weird cases like Qualith.  I may simply have missed some.
   1. Pseudo-Languages (not covered by anything granting language proficiency unless stated otherwise):
      1. "Telepathy" when it overcomes linguistic barriers - for example, both Mind Flayers and Myconid Sovereigns (as well as adults and sprouts) have telepathy, with different rules.  Mind Flayers have it in their languages list, so it uses the MM definition, whereby the user can communicate with any creature that has a language.  This means a Mind Flayer can't telepathically engage with a Myconid Sovereign, who has no language.  However, the Sovereign version isn't in the languages list and obeys its own rules, which means the Sovereign can engage with a Mind Flayer telepathically (via its Rapport Spores).  Telepathy is not generally selectable via a language proficiency.
      2. Qualith, as I discussed above, from Mind Flayers.  This is probably covered by Comprehend Languages, per its specific rules.
      3. There are a variety of ways to be able to communicate with "animals", typically a specific subset of them.  For example, Bullywugs, Gnomes, and Tritons all have this. 
   2. Racial Languages not in the dndbeyond list (I'm listing them under primary speakers, not other creatures that also speak them); by definition, these languages only exist in your setting if the race (generally species) that speaks it exists in your setting:
      1. Aberrations
         1. Grell
         2. Ixitxachitl
         3. Otyugh
         4. Quori
         5. Slaad
      2. Beasts
         1. Giant Eagle
         2. Giant Elk
         3. Giant Owl
      3. Constructs
         1. Modron (spoken by the Modrons, constructs native to Mechanus, like the Monodrone)
      4. Fey
         1. Blink Dog
      5. Plants
         1. Vegepygmy
      6. Monstrosities
         1. Deep Crow
         2. Hook Horror
         3. Ice Toad
         4. Kruthik
         5. Sphinx
         6. Tlincalli
         7. Umber Hulk
         8. Winter Wolf
         9. Worg
         10. Yikaria (spoken by Yakfolk)
         11. Yeti
      7. Humanoids
         
         1. Aarakocra
         2. Bullywug
         3. Gith
         4. Gnoll
         5. Grung
         6. Leonin
         7. Loxodon
         8. Minotaur
         9. Sahuagin
         10. Thri-kreen
         11. Troglodyte
         12. Vedalken
   3. Regional Languages:  Not every setting simply has Common and Undercommon as the universal languages spoken by various humanoid cultures.  This list is infinitely large, of course, but here are examples from released modules, all from Faerun:
      1. Bothii (but a separate module implies Reghedjic doesn't exist on Faerun in 5th)
      2. Netherese
      3. Thayan

Quote from Garret_Underbough >>

I am making a warlock and am faced with one of the most overwhelming parts of character creation: languages. I have searched everywhere, but I can’t find a list with every option. What are all the languages? Who speaks them?

 
It may not help to look at things that you may not be able to have.

.../phb/personality-and-background#Languages says that:

Your race indicates the languages your character can speak by default, and your background might give you access to one or more additional languages of your choice. Note these languages on your character sheet.
Choose your languages from the Standard Languages table, or choose one that is common in your campaign. With your DM’s permission, you can instead choose a language from the Exotic Languages table or a secret language, such as thieves’ cant or the tongue of druids.

How do I add languages to my character on dnd beyond??

Quote from ShynonieRed >>

How do I add languages to my character on dnd beyond??

As you build your character initially you will be able enter the starter languages and any they learn as class features or feats.
On top of that you can click on where it says "proficiancies and languages" on the bottom left of the character sheet a pop up of your proficiencies will appear on the right at the top of which is drop down list labelled "choose an option" from there you can either select an existing language (which includes world specific options like Zemnium) or custom which allows you to write in anything

This is very helpful to me thank you for posting this and all the research.