Several My Hero Academia characters have been Quirkless at some point in their lives. Having a Quirk is a significant deal in My Hero Academia, those without one are basically seen as lesser beings and are a minority in the world. For some of them, they were born without a Quirk and would later receive one via one method or another.
For others, they were born with one before being unable to ever use it again. Unsurprisingly, this also means that there are plenty of Quirkless characters in this series.
Note: There will be some manga spoilers.
4 My Hero Academia characters who were born Quirkless
4) Izuku Midoriya
Midoriya being born Quirkless was an integral part of his character at the beginning of My Hero Academia. He desperately wanted to be a hero, but his lack of abilities made him nigh useless and a common target for bullies to pick on. It wouldn't be until Midoriya tries to save Bakugo from the Sludge Villain that All Might is impressed by his heroic nature.
That would eventually lead All Might to transfer One For All to Midoriya. Technically, Midoriya doesn't just have one Quirk, he has six other Quirks at his disposal. In a way, he genuinely went from zero to hero.
He is the main protagonist of My Hero Academia, so it makes sense that he has so many powers that other characters could only dream of having.
3) Melissa Shield
Unlike every other character on this list, Melissa Shield never got a Quirk; she's been Quirkless her whole life. Even so, she still had a prominent role as one of the main characters in one of the movies (My Hero Academia: Two Heroes).
Although she could never become a traditional superhero, she is still able to become a valuable ally by creating inventions to assist heroes.
For example, she created the Full Gauntlet, which allowed Midoriya to use One For All at its full power up to three times. She hasn't been relevant since the movie, but it's likely that she is still using her brilliance in inventing new gadgets to assist heroes regularly.
2) Yuga Aoyama
One of the most shocking parts of My Hero Academia was that one of U.A.'s students was a spy for All For One. Many people speculated that it was Toru Hagakure, but it was later revealed to be Yuga Aoyama. The reason he spied for All For One was that he was born Quirkless and received the Naval Laser Quirk from him.
Yuga and his parents fear that All For One will kill them if they don't cooperate with his plans. He would later betray All For One and align himself with the heroes once more, still with his interesting Naval Laser Quirk.
1) All Might
Surprisingly, Toshinori Yagi (better known as All Might) was born Quirkless in My Hero Academia. Like Midoriya, he would later obtain One For All from somebody else (Nana Shimura). He wanted to be a symbol of peace, which was enough for Nana to take him in and start training him with this newfound power.
All Might became Japan's greatest hero and defeated numerous villains during his reign at the top. He possessed One For All for nearly 40 years, transferred some of its powers to Midoriya and continued to be the number one hero for a while. It wouldn't be until his bout with All For One that the last ember was finally extinguished.
Since then, he was Quirkless (although that doesn't necessarily mean he was no longer relevant to My Hero Academia's story).
4 My Hero Academia characters who could no longer use their Quirks
4) Mirio Togata
Mirio was born with the Permutation Quirk, which allows him to move through solid matter. It's a valuable ability to have, and his extensive training allowed him to master it. My Hero Academia fans should recall that he became Quirkless when he was shot by a Quirk-Destroying Bullet when he was trying to protect Eri.
Fortunately for him, this setback was only temporary; he would later regain control of Permutation when Eri (through some extensive training of her own) used Rewind on him. Still, that happened in the Paranormal Liberation War arc, meaning he went several arcs being Quirkless.
3) Cathleen Bate
She's also known as Star and Stripe and was the number one hero in the United States of America. Her Quirk was New Order, which essentially allowed her to create a rule that the target must abide by (with some limitations). It was a broken ability, so it was no surprise that All For One wanted her Quirk.
Unfortunately for her, she would eventually face Shigaraki in a match, where he would ultimately steal her Quirk from her in My Hero Academia. Fortunately for everybody but the villains, she managed to use New Order to destroy itself.
She was Quirkless shortly before her death, which predictably means that she could never get her Quirk back.
2) Tomoko Shiretoko
The Wild, Wild Pussycats are a fun group of characters in My Hero Academia. One of its members, Tomoko Shiretoko (also known as Ragdoll), was kidnapped. Her teammates found her after she was in the Hideout Raid Arc, which is when My Hero Academia fans discovered that All For One stole her Quirk.
Her original Quirk was Search, which allowed her to observe up to 100 people and identify their weak points (while permanently storing this information). It was incredibly useful and was eventually transferred to Shigaraki.
Since then, poor Ragdoll has been Quirkless and cannot help her teammates anymore.
1) Kai Chisaki
He's technically not Quirkless, but his Quirk requires him to use his hands. Considering that Mr. Compress and Shigaraki destroyed both his arms at the end of the Shie Hassaikai Arc, he can now no longer activate Overhaul.
That ability allowed him to reassemble matter with his hands, and it's impossible for him to do that when he has no arms (which means he has no hands by default).
This is his Quirk factor, which means that other My Hero Academia characters could potentially be unable to use their Quirk if the activation requirement is impossible. If he ever somehow got his hands back, then he could reuse Overhaul.