We have a large fishing club that is a 501(c)(7) organization. They run a very large tournament each year and collect entry fees from non-members. Could this tournament jeopardize their tax-exempt status since 501(c)(7) organizations may receive only up to 35% of its gross receipts from sources outside of its membership.

Yes, it can be an issue. The majority of income must come from members through dues and fees. A nominal amount may be collected as fees from non-members, but as you note if the anmount exceeds 35% the club may have its tax-exempt status challenged. But the rules are not hard and fast - if the fishing tournament furthers the clubs normal operations (namly, fishing) then the IRS will be more lenient. But if the income was from activities not related to fishing (such as parking fees, food concessions, raffles, etc) it could be an issue. To avoid uncertainty the 35% rule is anice "safe haven" to stay within.