Yes, you can automate this with Acrobat JavaScript. In a way this is a simple script, but there is a lot around it that is more complicated.
Basically the script needs to
1) Add add fields to the PDF page
2) put data(name and date) into those fields.
The slightly difficult part is determining the location of the field. If it's the same location every time, then you manually place a field and then get it's position from the field's properties dialog.
The other slight difficulty is creating the UI for the automation script. There are three options
1) ToolButton - use the "app.addToolButton()" function
2) Menu item - article below.
3) Command, commands are created from the Action Wizard Panel.
Here are some articles on these topics:
Automating Acrobat can save you loads of time.
PDF Page Coordinates (page size, field placement, etc.)
https://acrobatusers.com/tutorials/add_custom_menu_items
Also, you should consider using a Stamp with your signature image as the signature.
Thom Parker - Software Developer at PDFScriptingUse the Acrobat JavaScript Reference early and often