How do Latter-day Saints reconcile Moroni 10:3-7 with John 20:24-29 and Mark 16:14?

Let us consider a continuum stretching between "no trust whatsoever" and "complete and total trust". We are all progressing (or regressing) along that continuum.

God is interested in our progression towards complete and total trust in Him, but He is not saying "get to the other end of the spectrum by next Tuesday or we're through". I suggest it is fair to say that He is more concerned with the direction in which we are heading than He is in our location today.

God wants us to develop trust in Him, and He gives us the opportunity to do so "line upon line, precept upon precept, here a little and there a little".

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Asking & Acting

Jesus' rebukes noted in the OP are not commands to never ask questions again, or to cease all learning, but reprimands for failing to trust commensurate with the evidence that had been given.

Thomas (and the other apostles) had been given reason to believe—he had felt the power of Jesus’ teachings, witnessed miracles, listened to the predictions of Jesus’ death & resurrection, received the promise of the Holy Ghost, and heard the testimony of trusted peers that Jesus had risen. Apparently God expects many people to believe—sufficiently to act—with less evidence than this.

But it in no way follows from this that acting in faith and asking in faith are mutually exclusive. We can and should do both all the time. Jesus commanded people "learn of me", "search the scriptures", "seek and ye shall find", and so on. He clearly wants people to learn and grow, not plateau.

(that rhyme was accidental but I kind of like it)

The reprimand was not for seeking additional knowledge; the reprimand was for doubting and failing to act upon what they already had good reason to believe. Faith, as James taught (see James 2:18-20), is not simply belief. It is believing enough to act.

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Doubt

My read of the scriptures leads me to conclude that doubt is usually, if not always, a tool of the devil. It is the devil who wants us to believe we can never be what God intends us to be; it is the devil who wants us to question that what God offers could truly be better than what the world advertises.

And it is the devil who doubtless⁼⁾ inspired doubt & fear in the disciples as they experienced the worst Friday of their lives. He wanted them to give up and, despite every evidence God had given them to the contrary, lose their faith in Jesus the Messiah.

Asking sincere questions isn't wrong, and we're told to expect results (see Matthew 7:7). Doubting what God has already revealed (e.g. Matt. 16:16-17), on the other hand, is a regression along the continuum.

When God gives us light, knowledge, truth, and abilities, He expects us to use them. And where more is given, more is expected (see Luke 12:47-48, Doctrine & Covenants 82:3). Clearly Jesus had high expectations of the apostles, and they were expected to not fall for the diabolical sermons of doubt hurled about by the chief priests (e.g. Matt. 27:42).

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Specific Scriptural Applications

In Alma 32, Alma gives his great parable comparing the word to a seed, and proposes one of the more famous experiments of the Book of Mormon--how to know if the seed is good (see Alma 32:28-43). The similarities between this experiment and God's pronouncement through Malachi "prove me now..." are clear. God expects us to put His promises to the test.

In comparing the growth of our faith to the growth of a tree, Alma is clear that this is a process. And he points out that it is possible for one to have knowledge in one thing without having knowledge of all things. After one has demonstrated that the seed is good, he poses an interesting question:

34 And now, behold, is your knowledge perfect? Yea, your knowledge is perfect in that thing, and your faith is dormant; and this because you know, for ye know that the word hath swelled your souls, and ye also know that it hath sprouted up, that your understanding doth begin to be enlightened, and your mind doth begin to expand.

35 O then, is not this real? I say unto you, Yea, because it is light; and whatsoever is light, is good, because it is discernible, therefore ye must know that it is good; and now behold, after ye have tasted this light is your knowledge perfect?

36 Behold I say unto you, Nay; neither must ye lay aside your faith, for ye have only exercised your faith to plant the seed that ye might try the experiment to know if the seed was good.

We may have sure knowledge in one particular thing, but this in no way absolves us of the need to learn more, nor does it discount the importance of faith. God has much, much, much more in mind for us than just convincing us that He is real.

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In Doctrine & Covenants 131:6 we read:

It is impossible for a man to be saved in ignorance.

The context is important and the theology in this section is deep, but for purposes of this post I'll just acknowledge one detail: knowledge of God is an essential part of the plan of salvation (I deliberately worded that sentence ambiguously enough to mean 2 different things, because both are true). See also John 17:3 for the importance God places on this knowledge.

We are expected to ask, learn, test, and so on. But we are also expected to act in accordance with what we've already been given.

• It is good to believe based on evidence. It is good to put God’s promises to the test.
• It is bad to tell God we don’t like the evidence He’s already given us.

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Conclusion

The 3 & 8 witnesses to the Book of Mormon were cited in the OP. Their testimony does give a reason to believe the Book of Mormon...but it certainly isn't the only reason. If the testimony of the witnesses disarms our disbelief just enough to read the book and honestly consider its invitations, I would consider that an epistemological success.

Should Thomas' prior experiences, plus the testimonies of associates, been enough to disarm his disbelief? Probably...but I wasn't there and I'm not qualified to judge Thomas.

But the degree of personal conviction that will weather any storm, and ultimately lead one, prepared, to the throne of God, does not come solely from believing the words of Mary or of Martin Harris. The knowledge between here and eternal life, including what is described in John 17:3 and Doctrine & Covenants 131:6, is knowledge that comes from God.

Is it okay to seek personal confirmation (Moroni 10:3-7), or should one just be content with what others have already witnessed and testified to (John 20:24-29, Mark 16:14)?

I conclude that it’s okay to ask for more information, provided we are also willing to act on the information we already have, even if the requested information does not come as quickly as we want it to.