Are there any free online games where you can customise your character and you don't have to download?
Maybe you'll try https://slotuniverses.com/free-slots/
It depends on what you mean by character customization. If you mean to change the skin by buying it on csgo trading sites, then counter-strike may be suitable for you. If you mean changing your appearance and facial features, then you should play elder scrolls games. By the way, ESO is now popular.