The first sentence is correct, although Native English Speakers would probably use a pronoun before "school" (for instance: a school, the school). The second sentence is not correct English grammar. To make the second sentence correct, you can remove the word "do." "There is no excuse for the universities to not use their resource." (although "resources" sounds more natural.) BUT Some teachers might say it is better to say "not to use" than to say "to not use." In "not to use," the infinitive "to use" is kept together. In "to not use," the infinitive is "split" by the word "not." There is a rule that says not to split an infinitive. But this rule is ignored in everyday English. And not every teacher insists on this rule. But to be careful, you could get in the habit of not splitting the infinitive, and then you would please everybody.

Hmm, i think the first is more formal or better english... both are totally accurate, however the second one appears to me as something that would be said by people that aren't super educated, and i imagine the first one would be said by someone who speaks very well.

I suggest the following phrases are better: There is no excuse for the school not to take advantage of the resources. There is no excuse for the universities to not use their resources. Hope that helps.