Can customers still apply a discount code to their order if they add a bundle to their cart?
If the customers qualify for the bundle discount, then they won't be able to use an additional discount code. This is a limitation posed by Shopify, as draft orders can't use discount codes.
However, if the customers don't qualify for the discount (e.g. they don't have all the products from the bundle in the cart) then they will go to a normal checkout and will be able to use a discount code.

To allow customers to combine (but not stack) discount codes in the checkout, you can simply turn on the option to apply bundle discounts with discount codes and also turn on the option to allow customers to stack bundle discounts with other discount codes. This can be turned on in Bundler -> General Settings -> Discount modes.

Another option would also be to turn on the option to only apply bundle discounts if the products were added to the cart from the bundle widget. You can turn this on in Bundler > General Settings > Discount modes.