I think ? there's a way in Photoshop or similar , of replacing the background with a fresh white one .
I guess you'd have to trace round if you will with the Marque Tool .
I agree with Floppy / Steve with getting the subject away from the background so there's less chance of putting a shadow on it .
Diffusing is a good idea - tracing paper , a clear plastic bag on the flash head is another BUT you have to remember that these things will be blocking or soaking up the flash light if you will , so you'd have to open the lens aperture a touch to compensate .
The reflector is a good idea as it would push if you like , the light back . Anything will do as a reflector , white bed clothes even ! .
The most important thing is to have the flash as far away from the lens axis as possible , you could put a old film flash on a bracket or use a hammerhead Metz / Sunpak on Auto as long as the flashes sensor is near to the subject .
These can be obtained quite cheaply second hand . For instance I've a Sunpak " Potato masher " which I bought from a car boot sale for £4 GBP .
What you have to check on Google is Flash Trigger voltage , some old warhorses like the Vivitar 283 have high trigger voltages , that may not have harmed a old manual film camera but would fry the electronics in a Digital camera .
For instance I use a Wein Safe Sync on my elderly Canon Powershot G2 - snappy / back up Digital to get the voltage down to less than 6 volts .
To show the importance of moving the subject away from the background - flash used was the Sunpak " Potato Masher " - the flash head was angled upright into a clear plastic bag - the top of the bag was the ceiling if you will , and the bag was the diffuser .