A person who is machmir not to carry even when there is an Eruv, should he make an exception to bring food to a sick home bound neighbor or rather let the neighbor fend for himself?

The best choice is to ask someone that does carry to take the food for him. There was an opinion given to carry it without stopping at all, however Horav Shlomo Miller Shlit’a did not accept this Heter. Giving it to a child that anyway carries in that area, the Rov maintains is also debatable since after all he is following your instructions, and you do not carry. However, the Rov added that if you give him some of the food he likes and is taking it to the neighbor also, it is permitted since now he is willingly doing it for his own benefit. (See Kuntress Hama’ayan V’ 3: P. 63 on an article written by the Rov regarding this issue).