Which one of the two sentences is grammatically correct and why?

The first is correct, grammatically.

..while the second is not

….but why

It seems to be missing a transitional verb.

Both are clumsy; the second is also wrong.

This is correct as far as it goes; we use the gerund (-ing) form of a verb when we talk about having trouble doing something.

A supplier who has chronic trouble fulfilling his obligations to a customer will quickly lose customers.




This is wrong. We do not use the base form of the verb in that place:

A supplier who has chronic trouble fulfill his obligations to a customer will quickly lose customers.



However, the repetition of 'customer/customers' is clumsy and it would be better to limit the use of 'customer' in this way:

A supplier who has chronic trouble fulfilling his obligations to customers will quickly lose them.



But there are still too many words.

A supplier who cannot fulfill his obligations to customers will quickly lose them.

Or (why bother with 'chronic'?)

A supplier who has trouble fulfilling his obligations will quickly lose customers.