I don’t know of a movie theater or concert venue that will let you bring in your own food.
However, if you have some kind of condition that requires you to have either this or that type of food –
I would think the vast majority of venues would still insist you not bring your own.

What they might do is require you to have a doctor’s note before you do.

Again, “might allow you to bring in your own with a doctor's note!”

We had that experience at the PNC Bank Arts Center some years back when my wife was bringing along a jug of ice water from home – which she needed in order to take her meds.
The security stopped us and didn’t allow us entry, saying, in effect, that in the absence of a doctor’s note, she’d have to buy water from them – which, if you’ve ever been to a show at one of these places, doesn’t come cheap.

No harm – no foul. We emptied the jug and bought water from the vendors there.
That’s the policy – no outside food or drink.

Fast forward to this past weekend when a 16 year old boy with Type 1 Diabetes wasn’t allowed to bring into the drive-in – according to this –

a backpack containing insulin, an EpiPen, a juice box and candy to eat in case his blood sugar drops suddenly.

Apparently he takes the backpack with him everywhere he goes, and according to his dad, has other conditions as well.

Ironically enough, the place is owned by a pediatrician – Dr. John Deleonardis – who refuses to make an exception for the boy.

According to nj1015.com – the doctor/owner states

the drive-in is dependent on concession stand sales and he has food available that Ben could purchase. The pediatrician is concerned with people taking advantage if he starts making exceptions for outside food. He does not plan to change the policy.

The website notes that concession stand sales have always played a large part in the theater’s survival. “The concession stand is where the drive-in covers operating expenses. In either your own drive-in experiences of wanting to create drive-in memories for your children or grandchildren remember one thing, “If you eat here, we’ll be here” reads a post included with the theater’s history.

Weird to me that the boy hadn’t come across a situation like this before.
Is this the first time he’s encountered this?

But be that as it may – I could appreciate the doctor’s policy. As he stated above, he must have other foods that the kid could have bought there if he needed to. I’ve read before where they offer healthy alternatives to the usual theater fare - all part of their marketing scheme to keep the place open.

So either you abide by the rule or sneak your own stuff in – like that hasn’t happened before.