Is there a phrase or expression to describe a person or group that claims all of the work/tasks but can't deliver because they are so oversubscribed?
If I am not wrong, then you have requested an expression for a ventripotent coveter(s) who avowed to discharge all the work they(he) voluntarily requested to accomplished. But, now they(he) are(is) sitting in the stockpile of work and can't deliver due to their inefficiency. I regret, that I don't have a word or a phrase for this situation. Though, I have suggested three idioms, if you may prefer.
NOTE: Initially, I thought-out a word Sisyphean and I thought to share it with you. According to Oxford dictionary, it means: Denoting a task that can never be completed and a person/group doing such task can be called Sisyphus. But unfortunately, Greek mythology does not allow us to use Sisyphus for the situation you have described. Sisyphean task, according to mythology is a repetitive and never ending task. But, somehow Oxford Dictionary uses Sisyphean in a liberal sense. So, if we go with the flow, then, Sisyphus can be used for the person who does a task that can never be completed. It doesn't say anything about the task was alloted to them or they(he) vowed to do it by themselves(himself). But, according to Greek mythology it was a curse by God to Sisyphus. So, yes it was a given repetitive task by God to Sisyphus.
I can recall few idioms for this situation. Those three idioms are:
1) bite off more than they can chew: (Source: theidioms.com )
- try to do more than one is able to do
- undertake a promise one cannot accomplish
- accomplish attempt to do something which is hardly achievable
- to start or promise something to do more than one can accomplish
2) eyes are bigger than one's stomach: (Source: Merriam-Webster)
(used to say) that someone has taken more food than he or she can possibly eat
3) thou art like the Epicure, whose belly is sooner filled than his eye:
Explained beautifully by John Lyly in the book Euphues: the Anatomy of Wit' and 'Euphues and His England
I would say that those people or groups have overburdened themselves:
: to place an excessive burden on
// She overburdened me with work.
// Why overburden yourself when people are offering to help?
Note that although I think this is the most natural word, it doesn't actually necessitate that the work can't be done because there's too much of it for the people to handle. To make that explicitly clear, you'd have to say something like:
They overburdened themselves to the point of not being able to complete their work.
As an expression, you could say that they have bitten off more than they can chew:
If you say that someone has bitten off more than they can chew, you mean that they are trying to do something which is too difficult for them.
He bought the old hotel but soon realized he had bitten off more than he could chew.
You state that they take all of the work. The person or group represents a "monopoly" in the trade of that service. They set the price and the schedule because they are apparently the only ones providing the service. A single person or a company can monopolize business services or goods.