If the power should go off, I want my sump pump switched to a power inverter combination. Is this possible?
In theory, yes, you could, however you'll need to pay very close attention to the current draw and locked rotor amperage of the sump pump motor. If you've got a couple marine type batteries hooked up in parallel that can handle a 7.5 A or so drain for the 15 seconds it may take to clear the bilge, then you should be in the clear. I don't think I'd consider it on a single battery unless I purchased a system specifically designed to work on one. If you're considering adding a backup capability to the pump you have, I'd strongly consider two marine batteries in parallel with a trickle charger.
With two good batteries, you could probably get through a few (pump) discharge cycles before you outran the trickle charger and depleted the batteries. Of course, this entirely depends on the rating of the batteries and the pump motor.
You could, optionally, just buy a new pump that comes with a manufacturer certified battery back up system, or a pump that can work on water pressure alone if the power fails. As long as your water pressure meets the requirements of the pump, I strongly recommend the latter.
It's neat to build this stuff yourself, but I'd go for a certified solution if you really want peace of mind.
If you are thinking of running your sump pump with a power inverter connected to your car it is possible. The limiting factor is the current draw of the pump and the output rating of the inverter. Another consideration is the durability of the inverter since most models aren't designed for continuous use.
Inverters on their own require allot of battery power (like Tim explained) An alternative solution that might cost less or the same but with much better outcome is getting a small petrol or diesel generator.
The benefits are they can pump out a constant wattage as long as you got fuel. If your batteries die out and there is still no electricity - you can't charge them.
Another advantage is that you can pull some wire and place a separate emergency socket in your house and get some lights going, keep the fridge from defrosting, watch tv? or whatever else.
Disadvantage is that the normal sized generator can be loud and need to be hooked into an exhaust if used in doors, like a basement or loft. But you can buy slightly more expensive silent generators that offer the same power but when they run you can hardly hear them and are much smaller.
It all depends on how much power the sump pump needs and any other uses you could use it for.
Sizing the Inverter
If you want to run an appliance like a sump pump from an inverter, you need to make sure that you get one that's large enough.
The pump will be rated according to how many amps it draws (probably in the range of 5 or 6, based on a quick internet check). However, when an AC motor starts it uses a much higher current, maybe 2x as much. Ideally you would measure this surge current amount and duration so that you know what you're working with, but you could assume double the current for several seconds. So if you have a 5 amp pump, lets assume its surge current is 10 amps.
Inverters are usually rated by watts, but you can compute the amps by dividing by 120V. Inverters usually have a continuous wattage rating and a higher surge rating. Unfortunately, cheaper inverters will only surge for a fraction of a second before cutting out, so you can't just assume that the surge rating of the inverter will be able to supply the surge current of the pump.
The only way to really be safe is to get an inverter whose continuous rating is greater than or equal to the surge current draw of the pump. So in this case yo'd be looking at a 1200 watt inverter (which may have a higher surge rating). [Some of the nicer brands of inverters will publish how long the surge rating can be used for, e.g. 0.5 seconds, 2 seconds, 60 seconds. If you know that your inverter has a long surge time, you may be able to get a smaller inverter].
Sizing the Battery
If this is just for a one-off emergency usage you may be able to run the inverter from a car battery, but you should at least have a sense of what you're getting into. To compute the battery runtime you need to know the amp draw on the battery. Note that the amps coming out of the battery will be at least 10x higher than the pump's amps, due to the much lower voltage. A 5 amp / 120V pump is using 600 watts, which is 50 amps at 12V. Assuming an inverter efficiency of 80% you're pulling around 62 amps from the battery. The capacity of car batteries varies based on model, but a good sized battery probably has 50 amp-hours of capacity (that's 50 amps for one hour, or 25 amps for 2 hours, or 10 amps for 5 hours). So you're looking at getting about 45 minutes of runtime until the battery is totally dead. (Of course it's not good to run a car battery all the way down, and then you won't be able to start your car, so it probably wouldn't be prudent to actually run it for 45 minutes).
If this is going to be a semi-regular event, you probably want to get a marine / deep cycle battery or two. Those have a higher capacity and will withstand discharges better.
Automatic Transfer: Most small inverters just have regular 120V outlets that you plug your load into. If you want the inverter to be hooked up full-time and transfer from mains power to battery power, that will require something more sophisticated and probably a professional installation.
Wire Size & Fuses: As you can tell from my rough example numbers, you're looking at 60+ amps from the battery, which is a LOT. Standard household romex / extension cords is not going to be safe for that amount of current. You must properly size the wire or you will have a fire.
In addition, you should have a fuse on your wire with a rating less than or equal to the wire capacity. The purpose of the fuse is to blow before something catches on fire. If you're regularly blowing fuses it means there's a dangerous problem with your setup.
DC Pumps: If you want to run a sump pump from a battery on a regular basis, you're probably better off just getting a 12V DC pump and avoiding the inverter. You can run those directly from a car battery.