We show how certain active transport processes in living cells can be modeled in terms of a directed search process with stochastic resetting and delays. Two particular examples are the motor-driven intracellular transport of vesicles to synaptic targets in the axons and dendrites of neurons, and the cytoneme-based transport of morphogen to target cells during embryonic development. In both cases, the restart of the search process following reset has a finite duration with two components: a finite return time and a refractory period. We use a probabilistic renewal method to explicitly calculate the splitting probabilities and conditional mean first passage times (MFPTs) for capture by a finite array of contiguous targets. We consider two different search scenarios: bounded search on the interval [0, L], where L is the length of the array, with a refractory boundary at x = 0 and a reflecting boundary at x = L (model A), and partially bounded search on the half-line (model B). In the latter case there is a non-zero probability of failure to find a target in the absence of resetting. We show that both models have the same splitting probabilities, and that increasing the resetting rate r increases (reduces) the splitting probability for proximal (distal) targets. On the other hand the MFPTs for model A are monotonically increasing functions of r, whereas the MFPTs of model B are non-monotonic with a minimum at an optimal resetting rate. We also formulate multiple rounds of search-and-capture events as a G/M/∞ queue and use this to calculate the steady-state accumulation of resources in the targets.