What is proper way to "pass around the hat" for donations from a distance?

You could use Venmo, Zell, PayPal etc. if you are talking about putting together a non-reoccurig Parting gift like a plaque (not a plague); a one time gift for birth of a child as an example; a Solatia type gift to offset expenses due to a death/serious injury or sympathy gesture; etc these are all fine. The wording of the message needs to be an opportunity not a direction. Example:

PFC Smith is leaving and we would like to present him with a (insert thing here). If you would like to donate, you may make a contribution via (payment system). Proposed donation is $5 per person, if interested.

If you do it by donation by occasion , vice a dedicated non appropriated fund (like a cup and flower fund) you don't need to be concerned with running a fund. I would recommend that a follow up message should follow the presentation to the donators that explains dispo of the donation minus the soldier recognized. Example: we collected $40 which was enough to purchase a plaque, receipt attached. The left over change was donated to (a charity like UNICEF, AER, etc)