I assume that if we had two people named Ben and Jerry who made ice cream together, then it would be Ben's and Jerry's ice cream, yes? Now if "Ben and Jerry" were the name of a company, then it would be grammatically correct to call it Ben & Jerry's ice cream, but are there any problems here with the name of the company in its current form in terms of English grammar?
Look at this website.
According to the website, Ben's and Jerry's ice creams would be used if Ben and Jerry had two different ice cream companies. Ben and Jerry's ice cream is used if they jointly own a company.
Actual usage by native English speakers doesn't always strictly follow this rule, but Ben and Jerry's ice cream is the most grammatically correct way to name the company.
The actual quote from the website says:
When you are showing possession with compounded nouns, the apostrophe's placement depends on whether the nouns are acting separately or together.
Miguel's and Cecilia's new cars are in the parking lot.
This means that each of them has at least one new car and that their ownership is a separate matter.
Miguel and Cecilia's new cars are in the parking lot.
This construction tells us that Miguel and Cecilia share ownership of these cars. The possessive (indicated by 's) belongs to the entire phrase, not just to Cecilia.
are there any problems here with the name of the company in its current form in terms of English grammar?
No. None whatsoever. You can call a company or a product whatever you want (within reason) and use whatever punctuation you feel is correct. I suppose the owners could have called it "Be'ns # Jerrys !ce cream" if they had wished.