You can create a Submit button in your PDF form & add an action to it to send the PDF via email, refer to this KB doc. for help Acrobat Help | Setting action buttons in PDF forms.
I have tried to use this method to submit PDF forms but it does not seem to work in Chrome. Do you have any suggestions on that?
Chrome comes with it's own PDF viewer, which means that what works and what does not work with PDF files in Chrome is up to the Chrome development team. Or, to put it in different words: If something does not work in Chrome, you will have to complain to the Chrome developers about that.
In general, the functionality documented in the Acrobat SDK works when a PDF file is viewed in an Adobe viewer - that can be either Adobe Acrobat, the free Adobe Reader, or the browser plug-in that gets installed by either one of these applications. If you open a PDF file in e.g. Internet Explorer, and you have Acrobat or the free Reader installed, that PDF file will be displayed using Adobe's PDF viewer plug-in. If you are using FireFox, you would have to turn off it's internal PDF viewer and configure it to use Adobe's plug-in. If you are using Microsoft Edge or Chrome, then there is unfortunately no way to configure these browsers to use the plug-in, but you can configure them to open PDF files in the standalone version of Acrobat or the free Reader, and in that case, the submit functionality would work again. Take a look here for more information about PDFs in browsers:
To make sure that your users are using the correct environment, you may want to instruct them to either configure their browsers correctly, or download the PDF file to their computers and then open it in either Acrobat or the free Reader.