DeepL literal translation :

De n’importe où… à n’importe qui.

would have been an extremely poor choice, as the meaning is more like "we do not care a lot of about who would actually get what was sent from whatever location".

De partout means "from everywhere". This is positive. There are no places from where the mail won’t be picked.

À vous risks not to be immediately understood, because it can also mean "your turn" or "belongs to you", instead of highlighting the final destination/recipient.

Jusqu’à vous (up to your place) is conveying the movement implied by the English "from – to": Anywhere → transit → Anyone.