You cannot upgrade a 32-bit OS to 64-bit without losing your applications. That change from 32 to 64-bit, or 64 to 32-bit requires a clean install as they use different drivers so you will need to erase the partition but back up your data beforehand. It also goes to say that the applications will have to be reinstalled from scratch.
If you have plans on upgrading your RAM to 4GB then you will be forced to do a clean install of Windows 7 64-bit. Don't worry, you can still use an upgrade Windows 7 disc to do a clean install. Read these:
http://www.mydigitallife.info/clean-install-windows-7-with-upgrade-media-and-product-key-on-formatted-or-empty-blank-hard-drive/
http://nwarwick.co.uk/wp/index.php/clean-install-windows-7-using-upgrade-media/
Another option is if you have your Vista installer disc for the 64-bit version, do a clean install of Vista 64-bit (you lose your applications) and then upgrade to Windows 7 64-bit. However, that would require more work in your case as you will be installing an OS twice.
Now assuming you won't be putting 4GB of RAM in the near future, you have these options:
A. Use the Windows 7 32-bit upgrade disc on your old hard drive's Vista 32. You won't lose your applications as they will be carried over. Once you have changed to Windows 7 32-bit, clone the hard drive to the new drive with this:
http://www.partition-tool.com/personal.htm
You will need to put the new drive on an external hard drive enclosure to do that. If you don't have an enclosure, back up the upgraded OS to a few DVDs with this:
http://www.todo-backup.com/products/home/free-backup-software.htm
Once you have the backup of the whole drive on discs, restore them to the new hard drive.
B. Clone the old drive with Vista 32 to the new drive with Partition Master, or ToDo Backup if you don't have an enclosure. After cloning the Vista 32 to the new drive, use the Windows 7 upgrade disc.
Meanwhile, I you were to ask me, I would prefer doing the clean install of Windows 7 on the new drive using the upgrade disc by following the instructions in the first two links above.