Vintage Vespa scooter repair is generally pretty straightforward once you have a proper repair manual and some experience going through the process. One of the easier tasks is to remove and reinstall the gas tank. Depending on the model Vespa you have, the gas tank can be handled in a few steps. Later models involve a few extra steps but the process is generally the same.
Make sure you have the proper gas tank for your scooter model. This will probably already be the case, unless you are building a Vespa scooter from parts. If not sure, reference the tank to Vespa manuals to be sure. Once verified, confirm the fuel tap and filter are installed on the bottom of the tank and that the tank lid is functional and can be closed properly. If you have a later model tank with an oil tank, confirm your oil tank has a lid and that the oil reservoir has the sight glass and oil tap installed as well. Install the tank/body rubber gasket around the tank up to the top rim so that it sits between the tank rim and body when put in the scooter.
Make sure the engine carburettor lid is removed so you can have access to the carburettor fuel intake later on. On later models, you will also have an oil tank and an oil intake in the carburettor. Early Vespas don't have this concern since the two-stroke oil is mixed with the gasoline in the fuel tank.
Install the fuel and oil lines by taking one end of each and pushing them onto the gas and oil taps. The taps will have a nozzle that can be inserted into the hose lines. When properly seated, take a banjo clamp, slide it up the hose line until is at the hose part covering the nozzle, and then tighten it. Using a flathead screwdriver, twist the banjo clamp tightener until it is properly tight. Don't over-tighten or you could cut into the hose and cause a leak later on.
Now that the tank is ready to install, go back to the Vespa scooter it will be inserted into and clear any items that could be obstacles. Make sure the wiring that runs underneath the tank is properly in place. Confirm you have the rubber doughnut for later model Vespas that use an oil sight glass, but don't install it in the body yet. Confirm you have the on/off lever for the tank fuel tap but also leave that uninstalled. Some taps use a long lever that is connected to the fuel tap on the tank. If you have this, than disregard needing an on/off lever.
Slowly hold the tank over the Vespa body cavity it will be inserted into. Take the three feet of fuel line already connected to the tank and insert the end through the body grommet where it will come out and go to the engine. This will be the longest hose. You will do the same for the oil line.
Once both lines are through the body, begin lowering the tank lining up the fuel rod coming from the fuel tap and the sight glass from the oil tank with the holes in the Vespa body. When fully installed, both items will stick out of the Vespa body. Be careful lowering the tank; you want a comfortable fit. If you have to force something, you're doing it wrong and something is jammed.
Continue to pull the fuel and oil lines through the body as more slack is made available with the lowered tank. Remove any slack in either line, but don't pull hard or you could pinch either hose and cause a blockage. When the tank is fully seated in the Vespa body cavity, the fuel rod should stick out of the front of the body underneath the seat area on the scooter.
On early scooter models, the rod will have its own handle. On later Vespa models the rod has no finished handle. Instead, for later models, install the fuel on/off switch over the unfinished rod and screw it secure with a Phillips screwdriver. Then, also for later models, take the sight glass rubber doughnut and slip it over the oil sight glass so that it doesn't rub against the scooter body.
Take the two hoses coming out of the scooter body and attach the fuel line to the carburettor intake and the oil line to the oil intake in the carburettor box. Then close up the carburettor box. Next, confirm the fuel tank is seated comfortably. Insert the four bolts that secure it to the body and tighten them down. Don't over tighten or you will strip the threads in the body. The front screws are also used to secure the scooter seat, so install that part as you finish tightening all the tank bolts.
Take your time when working on a scooter repair, particularly anything having to do with your fuel system and engine. Rushing the job will almost always result in your forgetting a step or putting a part in wrong. The problem will make itself apparent later when the scooter breaks down. Also expect to need to change your hoses every few years. Gasoline and oil dry out rubber hosing, causing it to harden and crack. That in turn results in leaks and fuel system failure.
Clean out your fuel tank before installing. Gunk, residue, rust and dirt can all travel down the fuel line and into the engine causing a breakdown later on. A common plug-up caused by tank installation will be an air bubble in the fuel line. This is caused when there is too much slack in the fuel line, causing a location where air can get stuck. Running the fuel and oil lines too tight can cause the hoses to get pinched, reducing fuel and oil flow. Both will cause engine problems at higher speeds. Additionally, over time the hoses can rub against a scooter part when tight and open a hole, leaking their fluids. Also make sure all your taps, nuts, and clamps are tight. Anything loose can result in a leak. Gasoline is notorious for finding openings and leaking out since the fluid runs much thinner than water.