The 5,000-signature document was taken to the Scottish Parliament.
Dumfries and Galloway Council wants to house the hoard in a new gallery being built in the region.
However, National Museums Scotland is also bidding to host the artefacts. A meeting takes place on Thursday to discuss their fate.
The objects were discovered by a metal detectorist in Dumfries and Galloway in 2014.
The Galloway Viking Hoard (GVH) group is backing the council's bid to ensure they end up in Kirkcudbright.
Cathy Agnew, who chairs the campaign, said: "The message from Galloway, Scotland and around the world is very clear - the hoard was buried in Galloway for safekeeping 1,000 years ago and that is where its home should be.
"We have huge support from the general public, academics, politicians of all parties and many others.
"It would be a travesty if their voices were ignored."
National Museums Scotland has said it believed it had put forward a "mutually-beneficial and positive proposal".
It would allow part of the hoard to go on display permanently in Kirkcudbright and the entire collection to be hosted, on occasions, by the gallery.
The Scottish Archaeological Finds Allocation Panel meets this week to discuss the hoard's future.
It will then make a recommendation to the Queen's and Lord Treasurer's Remembrancer on where the treasure should end up.
A petition has been delivered seeking to ensure a hoard of Viking treasure discovered in Galloway goes on display in Kirkcudbright.