This is an interesting idea, but far more complicated than it may sound on the surface. Here's how I'd approach it...
The world map is divided into regions to represent game areas. Each region has an encounter deck prepared for it. Custom quest cards would be provided for each of these decks which would contain victory conditions. These conditions would need to be met in order to pass or clear that region. The victory conditions should be based on the quest conditions for that encounter deck (since many encounter decks are designed around a specific challenge and would not function without these conditions). 3 levels of victory are possible, indicated on the quest cards.
Once a game has been won in a region, the player may move to an adjacent region.
For rewards and progression, I'd have players start with a core card pool (core set potentially) and upon entering a new region, ante up a random player card from the set appropriate to that region (heroes included). A level 1 victory (easy) lets you pass the region and pick up the ante card. A level 2 victory lets you keep the ante card and 1 random additional card. Level 3 gives you the ante card plus 1 random and 1 chosen card.
After a victory, remove 1 damage from each hero, remove all damage from allies.
A threat modifier is tracked throughout the campaign starting at 0: a level 1 victory gives +1, level 2 gives +0, level 3 gives -1. The modifier total is then added to your starting threat whenever you begin a game in a region. As with the core rules, threat of 50 loses the entire campaign.
Losing a game means losing a random card from your deck and a +1 threat modifier.
Safe regions like Rivendell or Lorien can let you trade cards from your pool for campaign level effects or special cards. You may trade X cards for a -X reduction of your threat modifier, or trade X cards to heal X damage from each hero. You may also trade a 3 cards for a boon or treasure card.
Extras: each region has a condition card which is considered in play during games in that region. these conditions may have different effects to reflect the flavor of the region. For example, in the Misty Mountains, eagle allies get +1 will. Or in Khazad Dum, the condition card can be exhausted to give a dwarf ally +1 defense until the end of the round.
An endgame would still be required, but this is how I'd probably approach the nuts and bolts of such a sandbox campaign.
This is probably full of design holes, but it's what comes to mind immediately.