If one were to insert the word "before" prior to "the Ark" then the answer is yes. They worshiped God while lying prone before the Ark. But the actual answer is no.
The question is worth asking, however. They did indeed bow down before a "graven image" of the cherubim and it is reasonable to wonder how this is not a violation of Ex. 20:3-5
You shall not make for yourself a graven image, or any likeness of
anything that is in heaven above, or that is in the earth beneath, or
that is in the water under the earth; you shall not bow down to them
or serve them...
Christians and Jews explain that although they bowed down before the Ark, and although priests did minister to it in various ways, this does not constitute either worshiping or serving the object itself. The problem here, of course, is that those whom the Bible considers as "idolaters" can make the same argument. For example, when the Taliban destroyed the huge, supposedly idolatrous statues of Buddhas in Afghanistan, the Washington Post reported that:
Buddhist teachers point out that despite the images' importance,
Buddhists do not worship them. "There is a misunderstanding that
Buddhists try to worship idols," said Guoyuan Fashi, abbot at the Chan
Meditation Center in New York City. "The main thing is that we respect
the Buddha because we understand his teachings."
Thus those who are considered idolatrous by Christians, Jews or Muslims, may understand themselves to be acting in a similar way to Joshua in the OP. Outwardly they seem to worship an idol but inwardly they are honoring their religion's teaching and striving to embody its principles.
Conclusion: Joshua and his companions did not worship the Ark, but the outward appearance of their act, in terms of body language, could easily be misunderstood as being little different from what the Bible calls idolatry.