Allow and allow of have essentially the same meaning, “permit”. They differ in the sorts of complements they license.
Allow licenses nominal phrases, gerund clauses (with optional possessive subjects), and infinitive clauses with objective subjects:
The evidence allows the possibility of an inside job.
The evidence allows (our) considering the possibility of an inside job.
The evidence allows us to consider the possibility of an inside job.
Allow of licenses nominal phrases and gerund clauses, but not infinitive clauses:
The evidence allows of the possibility of an inside job.
The evidence allows of (our) considering the possibility of an inside job.
Allow of has never been common, and it has practically disappeared today:
Allow for has a different meaning: it means to ‘take into account’ or ‘make provision for’. For instance, we might say:
A contingency fund in the budget allows for unexpected costs.
‘Fuzzy logic’ in the software allows for incomplete data.
We would not say that the evidence allows for a possibility, because evidence is not a viable subject for that action; it is rather your interpretation which must allow for possibilities:
A: We concluded that the safe was robbed by professionals, very likely the Simister Gang.
B: Did you allow for the possibility that it was inside job?
A: Yes, we took that into account; everyone with access to the safe had an unbreakable alibi.
Note, however, that these terms are so similar that they allow of easy confusion, so in your reading you must allow for the possibility that the writers have misused whichever terms they use.