Oh yes, yes, definitely! I did it twice!
Well, for my first birth, I did use the tub but only for about an hour at around 5-6 cm. I hated it, didn't get any relief at all in the tub, so I got out after an hour and did standing/rocking. That birth was completely natural, med-free, and went perfectly. 10-hr labor, 49 mins pushing, 9-lb baby. Never even had an iv port inserted, it was as natural as can be (for being in the hospital).
For my 2nd birth, I didn't use the tub at all. I presented to the hospital with PROM at 37 weeks; I'd given it 36 hrs after my PROM for labor to start, but I wasn't having contractions. My midwife was livid that I waited that long, and I was given the choice of being admitted and given pitocin, or waiting and being forced to do c-section (b/c they were all up in arms about possible infection). So I gave in to the pitocin, but I still did the entire labor/delivery completely natural other than the pit. No pain meds, none needed. No tub allowed due to my PROM (which I wasn't disappointed about due to my prior crappy experience in the tub). Once the pit kicked in, it was a fast, intense labor. 3.5 hrs long, 3 pushes, 7.5 lb baby.
So yes, you can certainly do it without a tub, especially with all the other helps you have there. You didn't mention a doula in your OP, so I'd recommend that, but you can do it with or without the tub or the doula (the doula just helps a lot!).