Are mission boards Biblical?

On the "biblical" question, may need to clarify.
"Biblical" as in "required by Scripture"? Certainly not.
"Biblical" as in "forbidden by Scripture"? I think this is also a pretty clear "not."
"Biblical" as in "allowed by Scripture"? This would be yes. There is even some precedent. That is, you do have more than one church involved. On one journey (first?) Paul is sent from Antioch I believe in Acts 13. But he is supported later by Philippi (says so in Philippians 4). Maybe not the same journey. Probably a stronger case can be made, but the idea of multiple churches working together to further the work is fairly evident in Acts.

Can a local church do what mission boards today are doing? This is where I'd suggest visiting or having a conversation or two with long-time reps of a few major boards. Talk to BMM and BWM, maybe GFA. There are lots of them.

Do some missionaries go out with the aid of a board who could go out with only the aid of their local church? Probably. I think it has much to do with where they're headed and how complex the process of getting and staying there is. But the teamwork boards facilitate on the field is often priceless as well.

I think churches can delegate some accountability, vetting etc to boards if they view the board as an extension of their own ministry as a church--and really behave that way (feeling some conviction on this point in regards to our own church just now!). If we're sharing a missionary, I don't know why we can't share--with other churches--the oversight. How to do that without using some third party that is neither my church alone nor any of the other churches alone? The board seems like a good concept for that.

I don't have really good answer on how these groups can be accountable/lead by local churches--other than to answer the question with a question. How did multiple churches cooperate in Acts? How did accountability work then? The answer is that you had apostolic oversight of multi-church efforts. So the "third party" (between churches or between a church and the missionary) was an apostle or the apostles collectively.
So what happens when they all die off? If you believe in apostolic succession or even that their authority continues in the true church as a whole, the answer is that your denomination takes on that role.
So if you believe in local church autonomy what do you do?

It seems to me you have three choices:
a. Do missions with no third party at all and no interchurch cooperation.
b. Do missions via a body that has some authority but intentionally seeks to honor the primacy of the local church
c. I forgot that one... it'll come back later. Maybe.

So which is better, no cooperation among churches or cooperation via an awkward middle entity?

As for lack of accountability, there are some ways to mitigate that. For one thing, I don't know of any mission boards that could continue to exist without a good bit of financial support from participating churches. So there's always the "Listen to us because we pay your bills" kind of accountability.
Secondly, I don't know much about how these groups are structured, but I can't think of any reason why the board that operates the ministry could not be composed of all or quite a few of the pastors (or other leaders from) the participating churches.

I do know that the boards I'm familiar with have no desire to usurp the role of the local church, but want to help in their efforts.
Maybe we can get one or two of them to comment here or post some links to their defining documents.

But I do think the ideal is for a missionary to be supported by a relatively small number of churches with whom they maintain very close connections. But the expense involved in getting to some fields just makes that very impractical.... and 2Cor. 1.11 suggests perhaps the more partners the better.

I don't think there needs to be a third part at all. I acknowledge the presence and position of apostles in the NT. However, you are still dealing with actual churches cooperating. I don't think we need apostles today to cooperate; each church still has an under-shepherd and flock making and implementing decisions to perform the work of the church as described in Scripture. What I am arguing against is non-churches taking over the place of churches. The church has been left to do a certain body of work. The church is indispensable to the age. When usurpers who are not churches start taking over the functions of the churches, I believe we have run afoul of Scripture.

I have no problem with legitimate parachurch ministries. These are folks who recognize the role of the church and seek to aid the church in fulfilling her own work. If mission boards want to help missionaries get visas and group health insurance, great! But when they become the primary leadership in vetting, discipling, sending, disciplining, monitoring, etc. missionaries they have crossed into the work of the church. This is happening all over the place, not just in missions. The church gets sloppy about her duty and someone steps in to "help." Think counseling, discipleship(schools and colleges), evangelism, pretty much every area of church life at this point.

Thanks for that, David.

There's always tension between what would be ideal and what we can actually do. I supposed in an ideal world, every local church would be well equipped to do all the vetting, training, etc.
And in an ideal world, the mission board entity would never overstep its bounds in serving local churches.
But reality is that sometimes churches can't do it and still want to be involved in missions. And reality is also that since we don't have apostolic oversight, sometimes para-church entities mess up.
(Even with apostles on the scene, Paul still had to write 1 Corinthians, had to confront Peter, had to help the Jerusalem council come down on the right side of Gentile regulation, so even if we had apostles, we'd be saying "in an ideal world, apostles would never have to..." )
Nothing works perfectly here below... yet.