Hi Arella,
Unfortunately, you won't be able to send the In-Browser Message Email Capture Forms without having the Braze SDK integrated. You mentioned you "do not have the Braze web API set up at this time" which is different than having the SDK integrated; can you please confirm?
I will say the Email Capture form isn't as straightforward on Braze as it is on Mailchimp; however, if your team has a Zapier account or is willing to purchase a Zapier subscription, I can definitely provide some guidance on how to get this setup. Zapier is an automation tool that will definitely make this process easier.
Let me know if you have additional questions!