The Bible contains sufficient enough indication of apostolic succession (though probably not "explicit" enough by unbiblical sola Scriptura standards to convince most Protestants: what else is new?):
St. Paul appears to be passing his office along to Timothy (1 Tim 6:20; 2 Tim 1:6, 13-14, 2:1-2, 4:1-6). See, for example:
2 Timothy 2:1-2 (RSV) You then, my son, be strong in the grace that is in Christ Jesus,  and what you have heard from me before many witnesses entrust to faithful men who will be able to teach others also.
There are many indirect indications. When Jesus gives His disciples charge to do certain things, it is seen, by and large, by Protestants, as commands to their successors as well (perhaps not always apostolic succession per se, but at least succession as believers in Christ). So, for example, when Jesus tells His disciples to preach the gospel or to baptize, virtually all Christians today think that this applies to all Christians in perpetuity. Yet when Jesus tells the same disciples to "bind and loose" (Matt 18:18; Jn 20:23; also to St. Peter individually in Matt 16:19), somehow that is not seen as a thing that is perpetually relevant through history, and is relegated to their time only.
This makes no sense. For one to take such a position, they have to establish a solid reason why they regard one instance as perpetual and the other as temporary. I contend that it can't be done; that any such criterion would be completely arbitrary. Often, sadly, it comes down to merely a contra-Catholic mentality and rationale: "Catholics believe thus-and-so, and so we must oppose it, no matter what the Bible may state on the subject."