This is legal English, which is a specialized field. If it relates to an important matter, you need to consult a lawyer. We are not lawyers.

I'm wondering whether the part marked in red is grammatically correct or not.

I'm also wondering the meaning of the words "the sums due." The amounts that need to be paid.

If anybody has knowledge of this, would you comment some here for me?

It would be much apprecaited.

Thanks.

p.s.: Do the "as reasonably necessary" mean "as much as it is needed" or something else?

'As necessary means 'as much as  needed'.

'Reasonably' means you can only do what a reasonable person would do.

eg You can't walk into someone's office and say 'I want to look at every single piece of paper you have, and I want to look at all your records for the last 100 years'. 

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Each party has the right to examine relevant records of the other party as reasonably necessary to check the correctness of the account or to determine the sums due Customer thereunder. 

Best wishes, Clive