Do all kind of unicellular asexual reproductions let mutations happen, resulting in evolution?
It is not essentially asexual reproduction that causes mutations. Mutations can be caused because of errors in DNA replication which can happen during both mitosis and meiosis. 

Mutations can also be caused because of error-prone DNA repair mechanism. 

Other than these intrinsic factors there are physical and chemical mutagens that alter the DNA or accelerate the intrinsic mutagenesis mechanisms. 

These mechanisms are universally common for all living organisms (that we know of :P ), however, their tolerance to mutagens and rates of intrinsic mutagenesis may vary. 

Have a look at this post too.