The relay shouldn’t be difficult.
How you do it depends on what you have to work with. Do you have two pstn lines in your home? (Vonage counts as one, assuming you are using the typical Vonage ATA device). Do you have a statis IP or at least a reasonably stable one?
If you have a stable or static IP, then you could register into your Asterisk server via SIP (assuming broadband connection is available). Then using an FXO port card hook up the Vonage line to your Asterisk server. Set the FXO port channel as your outgoing context, enable your SIP client as a friend who uses the outgoing context to dial out.
If your ip is dynamic and changes quite often - then being able to reliably hook up to it via SIP would require the services of a dynamic DNS provider - which is beyond the range of this forum’s interest and I couldn’t explain what they do in a hundred words or less.
If you have two pstn connections (including Vonage), then you can use two FXO ports - call in via one port, use a special extension that is set up to allow a dial out via the other (probably the Vonage port).
I haven’t done this myself, but if you look at the various extension examples you can envision how to use the IVR capability to (for example) once you are hooked up to the extension, offer the option to press 5 to make an outside call, if you press 5 it then waits for 10 dtmf digits and passes those appropriately filtered and parsed (dialing 1 or dropping the area code if local) into the outging context. This could be extension 8928 or something obscure like that where it’s unlikely that a random caller would stumble into it.