Convincing property owners to sell

Why are the two available plots not an option?

I also think it’s a good idea to offer the lady help. As has already been written here, mowing the lawn, shopping, carrying heavy things. Introduce yourself as nicely as possible, make a good impression so that the potential new neighbors appear in a good light. Apparently, you do have to convince the mother so that the daughters agree.
The strictly businesslike approach that suggests is probably not going to be well received.
Where do the daughters live, also locally? Or would they be happy about relief in everyday life and would encourage their mother?
Nevertheless, the prerequisites for construction must definitely be clarified before the purchase, that is very clear. The first step remains to convince the potential sellers.

The brutally matter-of-fact approach that @ypg suggests is probably not going to be well received.

But you don’t buy a property with offhand remarks and emotional sparring with someone who doesn’t own the property.

I also think it’s a good idea to offer the lady help. As has already been written here, mowing the lawn, shopping, carrying heavy things.

So examine carefully who you bind yourself to forever: what is that supposed to mean now? Are these supposed to become persuasion strategies, to sacrifice yourself to the neighbor for years? I am anyway of the opinion that one shows solidarity in a neighborhood community, which also includes shoveling snow sometimes if the other apparently can no longer do it. But in my eyes, those are no arguments that I should be sold the property, or the property at all. Facts count there. If you don’t even know whether a replacement garage has space "elsewhere," then a "don’t worry about it" doesn’t work. A site plan would also be extremely advantageous for discussion here. But I think the OP has not done her homework to be perceived as a potential buyer.

Maybe one should also be willing to offer a TOP price for THE SELLER when receiving something better than what is otherwise available.

But you don’t buy a property with casual remarks and emotional banter with someone who doesn’t own the property.

Maybe not with casual remarks, but possibly with emotions. Because it seems that the old lady has decision-making authority. Perhaps the daughters care deeply about her well-being. And she is the one who basically bears the consequences, the construction noise, the dirt, months without a garage. Who knows if the new neighbors are loud party animals? Therefore, it certainly makes sense to present yourself as good neighbors.

I would also find a site plan very useful. I especially cannot imagine the location of the garage and alternatives. If the plot is better than the other 2 alternatives, then I would also offer a correspondingly good price. I also believe that emotions will play the decisive role, but if someone offers me 200k, my emotions are better than at 100k...