No, it is not possible to automatically send a letter created in Page Designer. The only way to get a document out of Page Designer is manually in the user interface.
However, it is possible to automatically create .pdf files using other 3rd party services, upload the pdf file to an attachment field, and then email the attachment. There are various 3rd party services such as
Zapier & Google Docs
Depending on your exact workflow and the complexity of your document, it also possible to use a combination of custom scripts and 3rd party service that have lower monthly fees.