As Paul Dessert mentioned, part of it is a business decision. What are the consequences of NOT having email info for somebody? Would you still be able to contact them? Do you need to contact them in the future?
If the answer is no (e.g. you have alternate methods for contact), consider this approach
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This is the approach we've used in a cellphone service signup flow. A sales rep is suppose to ask for the customer's email info and only if they don't have one (e.g. grandma who's not tech savvy) would they use the checkbox "escape hatch".
I believe it should translate OK if the person is directly filling in the form themselves. Email is fairly commonplace nowadays. Putting the email input right next to the "Email" label ought to be enough to prompt people to enter their email address.
It doesn't hurt to test it to confirm there's no change in the behaviour of the rest of your users.
I want to also point out, just because you have a field, it doesn't mean the user has provided a valid/their own email address. So if you're doing the test, it might be a good idea to visually scan the list of emails inputted before and after the change to verify the quality of email inputs. It could be more people then expected are selecting the "don't have email address" checkbox, but it accounts for the bogus addresses you've been receiving previously because people have no other alternatives for continuing.