What is the difference between Can someone come to my home to pick it up? and Can anyone come to my home to pick it up? ?Feel free to just provide example sentences.

"Can someone (do something)?" is the normal way to say this sort of thing. It is just asking whether there is a person who can do that thing, or whether a business will make one of their people do that thing, etc.

"Can anyone (do something)?" implies that most people can't do that thing (it's an unusual ability, etc), and you're asking whether there is anyone present who has the ability to do it (or whether everyone here is unable to do that thing). For example, "Can anyone speak Japanese?" (I'm looking for someone who can speak Japanese, but I know most people (here) can't, and am not sure whether anyone with that ability exists, etc)

So usually, if you were, for example, wanting to mail a package and talking to the package delivery company, you would say "Can someone come to my home to pick it up?" to see if they offer that service. However, if you were talking with a group of people who you knew were all busy, but you're hoping one of them will have time to pick something up from your house, you might say "Can anyone come to my home to pick it up?", because you're really saying "I know most of you probably can't do it, but does anyone in the group have the ability to do this?"