There is growing interest in understanding and controlling the spread of diseases through realistically structured host populations. We investigate how network structures, ranging from circulant, through small-world networks, to random networks, and vaccination strategy and effort interact to influence the proportion of the population infected, the size and timing of the epidemic peak, and the duration of the epidemic. We found these three factors, and their higher-order interactions, significantly influenced epidemic development and extent. Increasing vaccination effort (from 0% to 90%) decreased the number of hosts infected while increasing network randomness worked to increase disease spread. On average, vaccinating hosts based on degree (hubs) resulted in the smallest epidemics while vaccinating hosts with the highest clustering coefficient resulted in the largest epidemics. In a targeted test of five vaccination strategies on a small-world network (probability of rewiring edges=5%) with 10% vaccination effort we found that vaccinating hosts preferentially with high-clustering coefficients (similar to real-world strategies) resulted in twice the number of hosts infected as random vaccinations and nearly a 30-fold higher number of cases than our strategy targeting hubs (highest degree hosts). Our model suggests how vaccinations might be implemented to minimize the extent of an epidemic (e.g., duration and total number infected) as well as the timing and number of hosts infected at a given time over a wide range of structured host networks.