Which should I use: allows not or does not allow?
Can I use both?
Are there verbs that does not allow the two forms?
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Allows not is archaic. There is also a small problem with word order and where the object of the verb goes. These days it is only used for particular effect.
There are only two verbs in modern English whose present tense is always negated by following with not: do and be.
He does not...
He is not...
Every other verb uses "do not" to negate it: "He does not allow..." as opposed to "He allows not". The second form is used only for effect.
As Barrie has commented, have is also a modal verb along with do and be, and it can also be negated by following it with not. There is some evidence* that generally it isn't nowadays when the verb has a stated object. Where the object is stated, it's usually treated the same as every other verb.
He has not.
He does not have it.
He has it not. [Not generally used]
* Because "Has not...?" is a reasonable way of constructing a question, the Ngram is not entirely conclusive.