Data scientists often draw on multiple relational data sources for analysis. A standard assumption in learning and approximate query answering is that the data is a uniform and independent sample of the underlying distribution. To avoid the cost of join and union, given a set of joins, we study the problem of obtaining a random sample from the union of joins without performing the full join and union. We present a general framework for random sampling over the set union of chain, acyclic, and cyclic joins, with sample uniformity and independence guarantees. We study the novel problem of union of joins size evaluation and propose two approximation methods based on histograms of columns and random walks on data. We propose an online union sampling framework that initializes with cheap-to-calculate parameter approximations and refines them on the fly during sampling. We evaluate our framework on workloads from the TPC-H benchmark and explore the trade-off of the accuracy of union approximation and sampling efficiency.