Hello there.
Since your Sa-HT220 is analog only, you'll have to use RCA cables to have sound. First of all, those your DishTV receiver have Yellow/white/red at the rear? If yes, then you simply connect an RCA (white/red) cable from the DishTV to the Panasonic receiver.
Does your Vizio have an audio output? If yes, connect the RCA there.
If it doesn't, and only has HDMI, then you have two things you can use:
1) an extractor
http/www.newegg.com/Product/Product.aspx?Item=9SIAA3X52P4773&cm_re=hdmi_extractor-_-9SIAA3X52P4773-_-Product
This item extracts the audio from the HDMI cable. Simply wire the DishTV through this, and then connect an RCA wire to your receiver.
2) If your Vizio has a digital audio out (optical), then you can use this:
http/www.newegg.com/Product/Product.aspx?Item=9SIA9083UC5291&cm_re=toslink_rca-_-9SIA9083UC5291-_-Product
This will take the audio from your Vizio, such as with apps or mirroring, and convert it into RCA so you can connect it to your receiver.
Just ask if you have any questions. ^^