In this paper, we focus on a multi-agent source seeking problem where the safety of agents is characterized by a set of linear constraints. In particular, the safety constraints are also dependent on the unknown environment states, which makes the source seeking problem challenging to solve. To overcome such a challenge, we introduce a new notion of measurable path and then specify the reachability condition for all agents. A time-sequence of exploration is further introduced to help the agents to escape the stuck positions. To provide a performance guarantee for our source seeking algorithm, we perform the regret analysis and show a sub-linear cumulative regret. Finally, we evaluate the effectiveness of our SafeSearch algorithm through a set of simulations.