How can I ask for recommendations from an acquaintance?

I don't know the culture in which you and your acquaintance live, but I would interpret all these actions precisely the opposite of how you are interpreting them.

If someone moved back to my city and wanted to have lunch, I would assume they wanted to be friends, or if we had parted on bad terms, to clear the air between us. I would expect the lunch to be about our relationship. Something as simple as "do you know a good roofer?" could be done over email, so I would expect the lunch to mean something more. If we had a nice chat, and happened to discuss roofers briefly at one point, I would still be filing all this under "becoming (returning to being) friends with this person who suggested that we lunch."

Then some time goes by and I haven't heard from my (new?) friend at all. I wonder how they are doing. Are they settling into the town? Friends think about each other. What did we talk about at lunch? Oh yes, the roofer thing. I can ask how that's going. "Did you ever find that roofer you were looking for?" that will be a good followup question to keep our friendship going.

So, for the question of "how do I ask people I don't want to be friends with to recommend a service to me?" I would say message them somehow. Email, FB, WA, whatever. Don't ask them to lunch (a friend thing) if you don't want to be friends. For the question of "how do I react to this person 'poking' me after not helping me?" I would encourage you to think positively about the interaction and appreciate that this person remembers what you asked. You can of course answer something like "no, I am still looking. I was confident that you might know someone having lived here longer than anyone else I know. Perhaps you have family or neighbours who've had work done?" in the hopes that you will eventually get a recommendation from this person. If you stay warm and friendly in your communications, it won't be seen as "poking" but as interacting with each other as acquaintances do.

It seems that you actually have two problems. One is about getting a recommendation, the other about having (or not) a relationship with person Y.

Y is probably annoyed that I poked him for getting some recommendation for contacts just because I have a problem and that I do not really care about a genuine relationship with him

From the entire description, it seems to me that person Y cares about the relationship with you, and his feelings were hurt when you did not keep in contact. As a result, he now seems to seek some "mild" revenge.

The only thing you can do is to get together with the said person, and explain about you leaving and about not being on contact. Of course, an apology will do a lot, even it will not be obvious in the first seconds.

I know this because I am this kind of person myself - I do not like communicating electronically / on paper... I prefer face-to-face. The result is that when I am away, I do not communicate.

I had / have a friend who was upset that I moved to another country and I was not able to help any longer. I explained to her that I did not move just to run away from her, and the kind of help she needed was possible to be provided using e-mail (it was about proof-reading some texts). I gave her my e-mail address, and there was a change during the same day to "test", that if she sends me an e-mail, I will answer. Overall, it seems that it worked. I proved I am still there when I am needed.

IMPORTANT: do not accuse people for getting angry. Ever. Especially when you are even remotely responsible for their anger, and when you care about them.

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Your second problem is about getting a recommendation. AFTER you discuss with the person and clarify that you did not have ill intentions by not communicating, you can jump to the discussion about recommendations. Make it sound like a curiosity why they poked you without any other reaction.

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Sometimes, when I ask something like

"Why ...?"

or

"What ...?"

I get circumspection:

"Why do you ask?"

So now I learned and I ask

I am just curious, why ...?

and most of the time I just get the answer, without other detours.