Can I install a finance program without admin rights?
Yes. In Money by Zaweba, Tools > Install for this user installs it for your account only, with a Start menu shortcut and no admin rights needed.

It can, without any fuss. Money by Zaweba installs itself for your Windows account alone, with no administrator rights. Use Tools, then Install for this user, and Money copies itself to a folder in your own user profile, adds a Start menu shortcut and links .ozm money files to the app, all without the admin prompt.
That suits family computers, where one person holds the admin password, and any PC where you'd simply rather not change things for everyone.
What the per-user install does
Installing for one user keeps everything inside your own part of the computer. The app files go to a per-user folder, so they don't touch Program Files. You get a Start menu shortcut to open Money like any other program. And .ozm files are associated with Money for your account, so double-clicking a money file in File Explorer opens it.
Other people who sign in to the same PC with their own accounts won't see your Money shortcut, and nothing about your install affects them. If they want Money as well, they can install it for themselves the same way, and each person's money files stay in their own Documents folder.
Doing it
Run Money, then choose Tools, then Install for this user. When it's done, close the copy you started from and open Money from the Start menu, so you're running the installed copy from then on.
If you prefer the command line, the same job can be done with the --install option, and --uninstall removes it again. That's useful if you set up computers for other people and want a quick, repeatable way to do it.
Your data stays separate from the program
Installing or removing the program doesn't touch your money files. Each money file is a single .ozm file, usually kept in your Documents folder, and backups are kept in a backups folder next to it. If you uninstall Money, those files stay put; install it again and they open as before.
That separation is also why a per-user install is enough. Money doesn't need system-wide components or a database service running in the background. The program reads and writes your money file, and that's all.
Privacy on a shared PC
A per-user install keeps the shortcut and file association to yourself, but anyone who can sign in as you, or who can browse your files with admin rights, could still find the money file. If that's a concern, give the file a password. Money encrypts it with SQLCipher 4 and AES-256, and without the right password it won't open. The dialog warns plainly that a lost password means lost data, so keep a note of it somewhere safe.
A good fit for a household computer
Put together, it's a tidy arrangement for a shared family PC. Each adult installs Money for themselves without bothering whoever holds the admin password. Each keeps their own money file, optionally locked. And nothing needs an internet connection or an account, since Money by Zaweba is free software under the EULA that works entirely offline.