Every email that you send out needs to have a "send from" and "reply to" email address. It is not CAN-SPAM Act compliant to send an email from emails that are not from the "originating domain." This means you need to use an email address from the organization you are sending from, not a 3rd party email provider like gmail, yahoo, aol, outlook, hotmail etc.
Here's an excerpt of what the CAN-SPAM Act says:
"Don’t use false or misleading header information. Your “From,” “To,” “Reply-To,” and routing information – including the originating domain name and email address – must be accurate and identify the person or business who initiated the message."
If you don't send from your own domain they won't get to the inbox as they'll be treated as spam. Email deliverability has a lot of rules and when you don't play within them, not only will your emails go through, you can also end up in violation of the CAN-SPAM Act.