Yes, it's ok for your appliances to connect to chassis GND. It's also ok to connect your chassis GND to shore power GND when you hook up.
Note that the inverter already connects (-) (battery/chassis GND) to its outlet GND. Check it with an ohmmeter. Thus, not only is the inverter outlet grounded, you don’t have an option for it not to be, because of the (-) wire being grounded. See below.
simulate this circuit – Schematic created using CircuitLab
When dealing with inverters you need to be careful about three things:
Make sure your inverter and shore power are isolated by a transfer switch. Never connect them together.
Make sure none of your loads bond GND to neutral, unless your inverter allows it.
Be realistic about the kinds of loads the inverter + battery can handle.
The reason is for not bonding neutral is, low-cost inverters are push-pull on both H and N legs. That is, H is 120VAC with respect to chassis GND, and N is also 120VAC to chassis GND. H and N are 180 degrees out, so they measure 240V with respect to each other. You can check this with a voltmeter.
More expensive inverters have a floating secondary, so these will allow a neutral-to-GND bond.
Let's talk about being realistic. In a van, why do you need an inverter, let alone 240V?
Most people want a fridge. But, you can get a 12V/240V type with a Danfoss compressor. It can automatically switch over when you connect shore power. They're surprisingly efficient - mine runs on one 40Ah battery with a 200W panel.
Laptops? These can use USB-C these days. Phone chargers? Plain old USB or USB-C.
Oven? Microwave? Toaster? Running these on battery+inverter would be expensive and dumb in a van; these can be shore-power only thing.
Blender or other small hand appliance? Sure, that might be worth an inverter. But I've also seen USB-C ones (like the BlendJet) that work on USB.
Satellite Set-top box + Flatscreen TV? Yes, these tend to want wall power.
Starlink? Also wall power.
Power tool chargers? Unfortunately, also wall power.
So for the couple of small exceptions you could make do with a fairly small inverter (1000W or less.)
For my pop-up I keep a small 400W inverter for those limited cases, like someone's old laptop. Since then I upgraded my truck, which comes with its own 100/400W convenience inverter outlet in the bed. Kinda lame, but could do in a pinch.