Can I hide the bundle from storefront search and search engines?

Yes, you can hide a bundle from storefront search and search engines. By default, bundles are linked to a display product. 

Just create a new metafield for the display product with the following attributes:

• "namespace" : "seo"
• "key" : "hidden"
• "value" : 1
• "type" : "number_integer"

If you are using a Shopify app to power your storefront search, it's subjected to the setting of the app. 

Reference: Manage SEO data with the Admin API