I'm blessed to be in a church which has had the same godly pastor for over a decade, but know of another church where the people suddenly find themselves without a pastor. What should they do? I've spent over fifty years in churches where the congregation or some sub-part of it chooses the pastor. Here's a distillation of what I've learned from seeing that process succeed and from seeing it fail. If you find it helpful, feel free to link to it or to reprint it.
Searching for a Pastor
For a church, having the right pastor is one of the Lord's most precious blessings. Having the wrong pastor can almost be worse than closing up. In a congregational form of church government, how should a church find a pastor?
By Depending on the Lord.
All of the best methods and search tools and consultants won't do as much good as simply trusting the Lord to send the right man. He's able to make the church willing to have the man and the man willing to have the church. He's able to bring them together. Depend on him. See Proverbs 16:3
When we turn from our own sins and from our own stubbornness and our own selfness and pride and immorality, we become people whom the Lord can guide more easily. We also become the kind of people whom a faithful pastor would love to shepherd. See Hebrews 13:17
The Lord is pleased to answer his people's prayers that he send workers into the harvest fields. He can move hearts. Prayer is more urgent than starting the search process. See Luke 10:2
By Knowing What a Real Pastor Looks Like.
A real pastor may or may not be young and handsome and charming with excellent manners and a compelling way of speaking. He may or may not have completed seminary training. He may or may not have a "successful" track record at other churches. Jonathan Edwards lived in the 1700's in Massachusetts. He is remembered as one of the most outstanding and faithful pastors that America ever produced, but the church where he was pastor the longest voted overwhelmingly to fire him. He continued faithful work at another church. To find out what a real pastor is like, read Paul's letters in the New Testament, especially those to Timothy and to Titus. See also Acts 6:2
By Welcoming the Man the Lord Sends.
He may not look very pastoral at all, and he may or may not arrive by expected means. He may just show up as a visitor at church some Sunday. Don't accept just anyone who arrives by such unexpected means, but don't reject God's man just because he arrived in an odd way. I was in a church one Sunday when the interim pastor asked a visiting Bible college student to give a testimony. That visitor spoke. He ended up being the pastor there for several years. In another church there was an urgent need for a music pastor. One of the members took on the task temporarily in addition to his regular full-time job. The church then hired a very polished fellow for the job. That man lasted about 6 weeks and left under a moral cloud. A few weeks later, the pastor announced that the man who'd done the job on a temporary basis was resigning his other job so he could be the full-time pastor of music and worship. The grateful congregation stood and applauded. The new music pastor is now in a different church but is still a faithful music pastor, nearly two decades later. See Hebrews 13:17; 1 Corinthians 16:10
By Seeking Godly, Wise Counsel.
Notice that the counsel should be both wise and godly. Some people (even some denominational leaders) have practical wisdom but lack godliness. Others are godly, but may not yet be very wise. That wise, godly counsel may come from inside or outside your own church; it may come from inside or outside your own denomination. See Proverbs 15:22; Psalm 1:1-3
By Listening to the Spirit.
If you're not at peace with the possibility of a certain person becoming the pastor, say so plainly before the decision is made. I once voted for a pastoral candidate even though I had mixed feelings and my wife thought he was the wrong man. He came. Eventually that pastor said, "If I'd known how much pain my coming here would cause, I never would have come." He was, indeed, the wrong man for that particular church. Once the decision is made, however, be careful not to cause divisions. See Jeremiah 7:23; Ephesians 4:15; Colossians 3:15
By Being Patient.
It's far better to wait for a year or two than to rush into accepting just anyone who comes along. See 1 Samuel 13:8-14
By Wanting the Lord's Glory, Not Yours.
The church will never again be "like it used to be" when pastor so-and-so was here. Each pastor's gifts are unique. A new man's style will sometimes grate on you. Still, you want a man whose passion is for the Lord's glory, not to build a huge congregation by being trendy. See Psalm 115:1
By Being Confident.
You can have full confidence that if you, as a church, sincerely seek God's man in God's way with faith, the Lord won't let you down. Decades from now, you or your children or people you don't yet know will be looking back, thankful for your faithfulness in seeking to find God's man for your church. See Isaiah 49:23