Is there anyway to prevent the site from showing how many points someone spent on a specific piece of gear? The last items received box for instance shows who won an item and for how much when you hover over the item. The guild leader wants to keep it hidden.

Yep. By going to Admin > Security > Feature Access, you can control who has access to the security control called "dkpmenu." This limits who has access to view DKP-related information. The points and whatnot will still show for those ranks that have access (at the very least, yours), and will be hidden from the others. The Last-items received box will only show "Received by XYZ" rather than "Received by ZYX for ABC points".

Will that also block everyone from seeing the current standings? I want people to be able to see what they have, but not necessarily what others have spent for gear.

I'm afraid so. It's an "all or nothing" configuration.

When someone can view their actual current standings, it's always more healthy to have them be able to audit their own information (or others, for that matter) to ensure it's correct. Otherwise, they could have accidentally been charged too much for an item (say and extra "0" in a typo), and never be able to request a correction.