Would a PC with no or limited ability to speak (like a character born deaf or mute) be able to use spells with Verbal components?
The rules for verbal components require “the chanting of mystic words. “ a person who cannot speak cannot do this.
However, if you want to play this kind of character, talk to your DM. So long as it doesn’t change the mechanics (e.g. you can still be heard and Silence or a gag still stops casting) how you flavor your character shouldn’t matter. For example, you could be dumb except when the magic takes your tongue.
Alternatively, a sorcerer with subtle spell metamagic could do this but that will burn through spell points pretty quickly.
This is theoretically possible, but be careful not to change the mechanics
The full description of verbal components for a spell is:
Most spells require the chanting of mystic words. The words themselves aren't the source of the spell's power; rather, the particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion. Thus, a character who is gagged or in an area of silence, such as one created by the silence spell, can't cast a spell with a verbal component.
Because the 2nd sentence clarifies that the words themselves aren't the important part, this doesn't completely rule out the possibility that a spellcaster could provide verbal components of a spell without actually using their voice by producing the required "combination of sounds with specific pitch and resonance" some other way.
However, if you decide to pursue this character concept, you should come up with an alternate way of producing verbal components that preserves the mechanics, in particular the fact that "a character who is gagged [...] can't cast a spell with a verbal component." For example, if you chose to produce the "verbal" components of your spell using a stringed instrument, it would change the mechanics of spellcasting, allowing you to cast verbal spells even while gagged (and conversely, it would prevent you from casting verbal-only spells if your hands are bound). On the other hand, if you chose to produce "verbal" components by whistling, that should give the same mechanics: you'll be unable to cast verbal spells while gagged or silenced. If you're a bard, you could choose an instrument that requires your mouth to play, like a flute. This will have the minor mechanical impact that you won't be able to cast verbal spells without your instrument in hand, but that should mostly not be an issue, since you'll be using this instrument as your spellcasting focus and therefore will be holding it most of the time anyway.
Alternatively, your DM might be fine with altering the mechanics of spellcasting for your character. In the stringed instrument example, you would effectively convert all verbal components into material and somatic components (material because the instrument is required, and somatic because it must be played to produce the required sounds). So ask your DM about your character concept, and they may allow it even if it deviates somewhat from the standard spellcasting mechanics.
The easiest way to do this without altering the balance or mechanics of the game, and not requiring any tool or instruments, would be to cast spells by whistling. This sounds like it'd be great flavour for a bard.