To my knowledge, we don't know the requirements beyond Alaya appearing and a person accepting the offer of her power or the ability to create/preform miracles in exchange for their soul(?) or afterlife as Archer put it it the ubw route.

There may be a way to summon or contact her directly but again to my knowledge we haven't be told about or shown it. Also considering she's taking people to become part of the counter force so they can prevent human extinction I'm assuming she has some requirements, otherwise all she cares about is the person being vulnerable enough or just in a position that when the offer is made they can't refuse and once they've sold their souls and they die she just throws them across time and space preventing human extinction.

Shirou(archer Emiya) he was given the offer so he could save a hundred people in front of him who would die otherwise, and as we know shirou's only desire was to be a hero and save people, the only other counter guardian we, or at least I, know of would be Okita alter who wasn't given the offer, due to Okita alter being born prematurely she would have died but apparently the prayers of her sisters reached Alaya who sustained her through life (I'm guessing) and made preparations to induct Okita into the counter force upon death

So I'm guessing Alaya can't normally take a person into the counter force, cause if she could why send Emiya when you have gilgamesh, why send Okita when you have Artoria, which mean the person needs to accept her offer or something from Alaya in Emiya's case the ability to preform miracles and in Okita's her life, for her to induct them into the counter force though Okita sort of pokes a whole in this since she while was given her life by alaya the offer would have been given to someone else, unless Alaya sustained okita till she was of a certain age then appeared and gave okita the offer which in that case it was litterally accept or die

But ultimately Alaya seems to be the ultimate necessary evil of the fate universe as she, from what we've seen, waits until a person is at their absolute lowest or in a position they can't reject her offer then sends them through time and space to keep humanity alive

While i don't condone her methods i can see it's to some degree necessary