Yes, you can. Although be aware that some things might not appear of the Disney website even though they have been booked e.g. Magical express (At least that's what the broker told me).
I am travelling to WDW for 11 nights next year during the start of the F&W festival. I decided to rent points at 9.5 months out. I contacted Davids first and they told me that they didn't have the points immediately available to book any of my choices. So I contacted DVC Rental Store as well who said they had points and availability for BWV preferred view available. I couldn't afford all 11 nights there, so I booked just the first 5 nights. The very next day Davids got hold of me to say that AKV points had come through (standard view at Jambo), so I snatched these up for the remaining 6 nights. I am so excited about my split stay - particularly since finding out that I was able to get a BoardWalk view studio at BWV. Upon payment and confirmation, I was immediately able to see both bookings on the Disney website.
The service I have recieved so far from both companies had been excellent. Contact with Davids company has been completely professional and comes with the reassurance that he has a good reputation and has been around for years. Meanwhile the woman I've been dealing with at DVC Rental Store has given me the friendliest and promptest service I've ever received from a travel company, and as someone who's travelled quite a bit, that's saying a lot. Obviously, I haven't seen the bookings through to completion yet, but so far I'm impressed.
I could of gone the private route and saved an extra few hundred dollars, but I'm far too much of a worrier to deal with that level of trust.