When reporting results of a bootstrapped independent samples t-test, can I just report the t statistic alongside the bootstrapped p value? Or do I also need to report confidence intervals?
You can do pretty much as you see fit to do.
Broadly speaking, it may be reasonable to either report or not report a confidence interval depending on what you want to achieve/convey.
If you really mean "what will some supervisor/colleague/journal editor/reviewer accept?", that's a very different question, which would depend on things we're not privy to (which supervisor or colleague? which journal? ... and what do their guidelines say?)
I'd certainly report both the statistic and the p-value, as you suggested (and probably something more conventional, such as an ordinary t-test p-value or a permutation test p-value, so people could see what difference it makes, if any).