Is it correct to say:
I will do the barbecue.
Let’s barbecue the vegetables.
Let’s grill the vegetables.
Let’s do some grilling.
Can we use do with barbecue?
Don’t worry I will do the barbecue.( and barbecue means vegetables etc.)
All of those sentences are acceptable standard English. But how frequently they get used in those particular forms is another matter. Quite rarely, I expect.
I’m sure lots of people have said, at one point or another, “I’ll do the barbecue”, but that’s not to say that it’s a widely used turn of phrase. And the other sentences sound a bit like something out of a children’s grammar book.
You seem to be asking several different questions. It's all very jumbled and it's hard to know what exactly you are asking about, because there is no precise context, yet again!
Please think very clearly what you are asking and give us context.
"Let's ..." is a way of suggesting an activity. "Let's" is an idiom. It's short for 'Let us' but 'us/'s' is never replaced by another pronoun.
"Let's have/do a barbecue!"
"Let's do some grilling!" does not mean the same as "Let's have a barbecue". I have no idea what it means or who would say it. A barbecue is a sort of 'grilling' in British English.
"I will" is a way of saying what you intend to do, or make an offer.
"I'll grill/do the vegetables, if you grill/do the meat"
"We'll have a barbecue!" is a way of saying you have decided what you and everybody's going to do.
A barbecue usually involves grilling meat and vegetables. Why talk about vegetables as a separate thing?
Unlike some people, I'm not worried about using 'do' in this context.
It's used whenever people are deciding who is going to prepare specific items for a meal to which everybody contributes.
"I'll bring/do a dessert."
"Mary's bringing/doing a salad."
Or if people are sharing the various preparation tasks.
"I'll peel/do the potatoes."
"I'll make/do the sauces."