Is it scriptural that all Christians should be looking for ministries?
Yes, it is absolutely scriptural. As Christians, we are each to be good stewards of all that God has given us and this includes our time, talents (spiritual gifts), treasures (finances), our temples (bodies) which are indwelt by the Spirit, and truth (the message of Christ). Being a good steward means ministry of various sorts. This would also include a person’s vocation or job. Viewed biblically, our jobs are ministries in themselves. An excellent book I like to recommend along these lines is, Your Work Matters to God, by Sherman and Hendricks.
Further, the Lord gives all believers at least one spiritual gift like teaching, or exhortation, or showing mercy, etc., and we need to be using these gifts in some way for his glory, for one day we are all going to be held accountable for our stewardships. Ministry can occur in many ways both in a church and outside the church in the community in which we live. The basic principle is simply this. Every believer is to be a minister, one who serves others in ministry. For more on this whole aspect, let me encourage you to download the ABCs for Christian Growthseries and check out Part 3. I think it will answer many of the issues you raised with scriptural support and explanation. If you have never gone through such a study such as the ABCs, it would help you in your walk with the Lord.
Related Topics: Basics for Christians