OK, first the good news, I just finished this whole process with Dell Vostro 100 with the same GPU (Dell part:
320-5631 : Integrated ATI Radeon X1150 Graphics, for Vostro 1000) as you so it is possible.
Now as you know, you can't do the 'in-place' upgrade using the upgrade icon in your system tray since you get the error regarding the GPU. I got the same error. As you will see below, I DID NOT get the GPU error when I did the upgrade via USB stick.
To be comprehensive, there are two general ways to go:
1. The OLD 'PRE-November' 'TWO STEP' process which involves doing the upgrade first (via USB in your case) and then an optional clean install after windows 10 is validated. This process was a two step process since the OLD 'fresh install' Windows 10 installer would not accept Windows 7 keys and would not validate unless window 10 had been previously installed.
2. The NEW 'November update' 'ONE STEP' process in which MS says with the new installer you can do a clean install and enter the windows 7 key and windows 10 will accept it.
Now, I did the FIRST (OLD) one so I can say it works (option two SHOULD work as well, but I didn't do it myself so I can't vouch 100% for it.)
Here's exactly what I did (the old way) on my Vostro 1000:
1. I had a license for windows 7 Home Premium
2. I used that license to install and validate the 32 bit version of windows 7 Home Premium
3. I tried to upgrade via the tray icon and could not due to the GPU error
4. I downloaded the Windows 10 Home 32 bit version using the MS Media Creation tool and installed it on a USB.
5. I booted in to Windows 7 and navigated to the USB and ran the setup application from there (UPGRADE approach)
6. The windows 10 upgrade process started and ran WITHOUT the GPU error
7. When it asked me for the key, I selected skip
8. When the install completed I checked the System page and it told me windows 10 was activated (since it must have found my valid 7 license)
(I could have stopped there with a 'upgraded' installation but I wanted a CLEAN install, so...
9. I then BOOTED from the USB stick, deleted all the existing partitions, and installed CLEAN (fresh.)
10. When it asked me for the key I choose skip
11. When the clean install was complete I checked the system page and windows 10 was activated (since windows 10 had been successfully installed via the preceding upgrade process.)
Now, as I see it you have three options for install:
1. Old way #1 - Do it exactly like I did (32bit all the way) except you would use the 32 bit PRO download to match your license
2. Old way #2 - Don't reinstall windows 7, (64 bit all the way) start at step 4 above. Just download and use the 64 bit Windows 10 Pro version to MATCH your existing Windows 7 Pro 64 bit install.
3. New way - Download the 64 bit Windows 10 Pro installer (to MATCH your license) and BOOT from the USB (fresh install.) When you get to the key entry DO enter your key since MS says the new installer will accept it.
The easiest way is obviously #3 and I would probably go that way and hope that MS is right and the new installer will accept your windows 7 key. If that does not work you'll have to reinstall windows 7 and do it the 'old' way as in options 1 or 2.
To summarize regarding the GPU issue: for whatever reason the GPU error does not stop the installation as long as one is upgrading or fresh installing from a USB stick. I know that makes no sense, but that's the way it is. Others have reported the same thing with regards to the Vostro 1000.
All the rest is just how to install windows 10 stuff.
I know the above is a lot of detail but I wanted to be precise, let me know if you have any questions.