Volunteers arrived at the church last Thursday morning to find stones and shattered glass on the floor.
Further investigations revealed the North Transept window had been damaged.
They were greeted with the same scene on Tuesday this week with further damage caused to the historic building
Simon Delaney, surveyor for the Beverley Minster Old Fund, said: "It is going to cost a couple of thousand of pound minimum just for the scaffolding alone and then of course it is the work that will go into fixing it.
"We will need to measure for the new glass and then that will need to go back in. The annoying thing is is we only had that particular window refitted a couple of years ago.
"It's going to be quite a challenge to get this all fixed, but thankfully no-one was hurt and thankfully it wasn't an elderly person's home targeted."
The crimes have been reported to Humberside Police in a bid to catch those responsible for causing the damage.
Mr Delaney has also issued a plea to those responsible.
He said: "Please think before you do anything like this. There are other things you could be doing with your time."