We study bootstrap percolation processes on random simplicial complexes of some fixed dimension d ≥ 3. Starting from a single simplex of dimension d, we build our complex dynamically in the following fashion. We introduce new vertices one by one, all equipped with a random weight from a fixed distribution µ. The newly arriving vertex selects an existing (d − 1)-dimensional face at random, with probability proportional to some positive and symmetric function f of the weights of its vertices, and attaches to it by forming a d-dimensional simplex. After a complex on n vertices is constructed, we infect every vertex independently at random with some probability p = p(n). Then, in consecutive rounds, we infect every healthy vertex the neighbourhood of which contains at least r disjoint (k − 1)-dimensional, fully infected faces. Using a reduction to the generalised Pólya urn schemes, we determine the value of critical probability p c = p c (n; µ, f ), such that if p ≫ p c then, with probability tending to 1 as n → ∞, the infection spreads to the whole vertex set of the complex, while if p ≪ p c then the infection process stops with healthy vertices remaining in the complex.