CiviCRM Core will (by default) send out an E-mail to Contacts who are assigned to an Activity.
You can change this setting to Not send out any E-mail to Contacts who are assigned to an Activity and handle Email notifications on a per webform level. If you need an example of that - let me know.
Here is an example:
On the form: E-mail + a checkbox (without ...
Yes (I think this should work for an Activity created via a webform as well?) - go to Administer -> Customize Data and Screens -> Display Preferences
"Do not notify assignees for" and select the Activities ...
You can specify which Types of Activities send emails via CiviCRM but setting it will affect Activities created both in CiviCRM directly and via webforms.
I can flesh this out if this solution would work.
In addition to JonG's answer, I would definitely recommend using a professional SMTP Gateway. For instance we pay 10€ / month for 10 000 emails with great deliverability. They will provide SPF and DKIM entries for your Domain and your emails will reach their recipients. In CiviCRM you just need to set the parameters and you're done!