Long story short I have ordered a new gaming laptop. I want to buy a game in the upcoming steam sales but not install it until I get my new laptop. Maybe 3-4 weeks from now. Will I be able to do this?
Yes, if you buy a game on your Steam account, that game is licensed on your Steam account for your use. You can then just log into your Steam account on any computer and install and play the game as required.
You do not need the Steam client installed to purchase a game but you do need a Steam account, then you can either purchase the game via the Steam website or download the client. You will need the Steam client to install and play the game, though.