No, You could say, "I need to get my tires filled ( with air )." You don't need to mention the word "air" since it is obvious that is what tires get filled with.

No, you would say "I need to put air in my tires"

I would vote for James' answer as the best. Pumping up sounds like you're using the old hand pumps and doing it yourself. No one's done that since 1920. But if that's UK speech I'm sure that's fine. Sounds a little off in the US but not impossible. Inflated is a bit stiff and too formal sounding. No one would say "inflated" in the US. Plus it suggests someone is putting the air in for you and that's not happening in the US. Could likewise be UK speech. And to say you have to get the tires "filled" also sounds strange. Filled? Yeah it's obviously air but filled normally refers to fluids. These are are intelligible but just not what the words we typically use in the US. Here it's just putting air in tires. Pump up I guess isn't really that bad. I might say "I gotta go pump up the tires in the car" and that's okay.

Another alternative is "I need to get my tires/tyres inflated." If you're getting your car serviced (like an oil change), you would say this to the service technician. Ex: "I'd like to get an oil change and my tires checked and inflated...if needed...please."

Scott's answer is fine but you could also say 'I need to pump up my tyres.' Best wishes Bob