Why would they?
You suggest that only a joint effort could dig it up, but not all branches of Christianity see any importance in finding such relics to begin with.
Protestants in particular attach little importance to such items. Yes God used the Ark of the Covenant as a physical sign to remind the people of spiritual realities, but God's presence is not bound to it. If it were to be found, it would have no magical powers or significance beyond that of a historical artifact. Admittedly it would be a very interesting artifact, but Protestants generally believe their time and money can be better invested in other pursuits.
In fact some Protestants would argue that it is better off not being found as it would only cause massive amounts of distraction. Since the spiritual realities it was made to represent have been manifest in other ways, it is better not to be confused. In particular the things (besides the Ark of the Covenant you could also lump in lots of other things related to the old temples) that were meant to point ahead are now not needed as we look back on the event they pointed towards.
Catholics and Orthodox both attach more significance to relics (although the details vary), but still they don't see them as changing the course of history. Even they do not view Christianity as somehow inherently lacking that finding the Ark of the Covenant would make it complete.
There are certainly many sub-groups of Christians that would be interested in this and that would even back such a venture if clues arose. In fact quite o few have over the years. Doctrinally speaking however these groups usually don't match up very closely to the larger branches they come from and so there is no particular reason to expect broader backing.
To my knowledge, many digs and searches have been undertaken, but they are all long shots because there are no particular clues to go on as to where it might be. Also it is small and would have been easily destroyed. Moreover it would be very difficult to positively identify even if found.
If some archaeologist hits the jackpot, then so be it. In the mean time the bulk of Christianity does not have a reason to make finding the Ark a big deal.