This paper considers an unsourced random access (URA) set-up equipped with a passive reconfigurable intelligent surface (RIS), where a massive number of unidentified users (only a small fraction of them being active at any given time) are connected to the base station (BS). We introduce a slotted coding scheme for which each active user chooses a slot at random for transmitting its signal, consisting of a pilot part and a randomly spread polar codeword. The proposed decoder operates in two phases. In the first phase, called the RIS configuration phase, the BS detects the transmitted pilots. The detected pilots are then utilized to estimate the corresponding users' channel state information, using which the BS suitably selects RIS phase shift employing the proposed RIS design algorithms. The proposed channel estimator offers the capability to obtain the channel coefficients of the users whose pilots interfere with each other without prior access to the list of transmitted pilots or the number of active users. In the second phase, called the data phase, transmitted messages of active users are decoded. Moreover, we establish an approximate achievability bound for the RIS-based URA scheme, providing a valuable benchmark. Computer simulations show that the proposed scheme outperforms the state-of-the-art for RIS-aided URA.