This work proposes a bootstrapping with positivity methodology to study random $U(N)^{D}$ invariant tensors in the large $N$ limit. As has been done for $U(N)$ invariant random matrices, we combine the Dyson-Schwinger equations and positivity constraints of moments to approximate the moments of such tensor models. As examples, we bootstrap the quartic and two hexic rank three tensor models. All models studied converge quickly, and for those which have known analytic formulae, they converge to such solutions. We conjecture new explicit formulae for all moments of the rank three quartic model and support this conjecture using bootstrapped results and explicit double-series computations with 'feyntensor'.