Can Travel-On exchange a ticket I purchased somewhere else?

No, unfortunately not. An airline ticket is connected to the channel through which you purchased it and no other channel is able to change the ticket in any way. In other words, if you purchased a ticket from Travel-On then only Travel-On and the airline can change that ticket; if you purchased a ticket from Travelocity, then only Travelocity and the airline can change it, and if you purchased from the airline directly then only the airline can change it, etc. As a courtesy, a travel management company, like Travel-On, can sometimes call the airline directly to assist with a change.