We had similar situation - our neighbours own the massive field directly in front of our house.
We were interested in buying, or renting, just a small part of it, specifically to put the kids trampoline in as our garden is so small.
My neighbour is friendly and we are on good terms, but she is very straight talking, so I asked her directly, in a conversation, if she would ever consider selling or renting part of the field and told her why we wanted it, to which she immediately replied 'no way'. She didn't want to relinquish her land really.
But then a few days later she came back and said 'I've been thinking, why not just put the trampoline there anyway?' So we did!
Obviously it isn't our field, but we lost nothing by asking. I think a formal letter may have alienated her a bit as we are on friendly speaking terms.
ON THE OTHER HAND - if we had written a proposal in a letter she may well have had time to mull it over and come to the same conclusion!
So I guess it depends on your relationship. If you are friendly, perhaps an initial conversation would work?