I would probably use the word diffident when someone is acquiescing to another's authority. The word connotes not just "simple" shyness, but also a lack of self-confidence and assertiveness.
1. lacking confidence in one's own ability, worth, or fitness; timid; shy.
Another option would be timid. It's a fairly common word and I don't think I need to define it here.
To more directly put across the point of submission to authority, you can consider submissive.
1. inclined or ready to submit or yield to the authority of another; unresistingly or humbly obedient:
2. marked by or indicating submission or an instance of yielding to the authority of another: a submissive reply.
Obedient is another (more positive) option. Again, a word in common usage, and I won't be defining it here (it's easy to look up).
The most negative options, generally used critically, are pushover and doormat. They are idiomatic nouns that describe a person who seriously lacks self-assertiveness and may be defined as follows:
- Slang One who submits meekly to domination or mistreatment by others.
- One that is easily defeated or taken advantage of.
In the same vein as the above two, there is another idiom, which I believe is peculiar to the US: milquetoast.
noun, (sometimes initial capital letter)
- a very timid, unassertive, spineless person, especially one who is easily dominated or intimidated:
a milquetoast who's afraid to ask for a raise.
And, as included in the definition of that word spineless is another negative adjective for someone lacking self-assertiveness. Again, it's a highly critical word. Given that the spine supports your body in an upright posture, I believe you should be able to see how the metaphorical meaning comes about.