According to their dictionary definitions the principal meaning (as verbs) would be to ask, to demand.
To requisition is a little bit more complicated (may take an infinitive) which is why I think, it is more likely to be used as a noun next to “make” i.e. “to make a requisition”.
As for the verbs to require and to request they seem to be closer and can be used to ask and demand something as follows:
I require a new car.
I request for a new car.
Which would be the nuance differences between them regarding the colloquial, formal or polite usage?
As a noun I know and can understand the differences:
A product request refers to its demand on the market while the product requirements refer to the conditions that the product must comply with.