To stay means “not to change location or value; to remain”. To stand means “to be in a vertical position (on one’s feet, in case you speak about a person)”. For example:
correct I must stay at home today.
wrong? I must stand at home today.
The latter sentence would imply that you cannot sit down at home today (perhaps because there are no chairs there), but this is probably not what you want to say. If you wanted to express that you can’t sit down, “stand” would be appropriate:
correct I have to stand for 8 hours a day because I work manually.
wrong I have to stay for 8 hours a day because I work manually.