As far as I understand the process, the Congregation for the Causes of Saints which took over part of the role of the Sacred Congregation of Rites when that august body was dissolved in the late 1960s, does require the "authentication" of some miracles in the name of a beatified person before they will recommend to the Pope that he should proceed to canonisation.
I don’t think time is a limiting factor. The miracles do not have to date back to the lifetime of the person in question. If, for example, you had prayed to N S for a cure for yourself or someone else, and that cure came about in a way that could not be accounted for by the medical profession, you could have this investigated as a possible miracle. Personally, I think that if you, as an atheist, made such a claim, it should be taken very seriously, but I’m not at all sure it works that way.
So, yes, modern prayers would count. As far as the procedure is concerned, I assume one would start at Parish level by presenting evidence to one’s Parish Priest, who would, I imagine, pass it on to his Bishop.