The RPCNA (exclusive-Psalmody/no instruments) most likely expects their churches to use one of the Psalters they made/make available (most recent edition: The Book of Psalms for Worship, 2010). But it could be no formal declaration is necessary.
The OPC/PCA joint venture publishing company, Great Commission Publications, produces the Trinity Hymnal, which is a semi-official songbook for these churches going back to the '60s (OPC) and '70s (PCA); but there is no denominational requirement to use it.
The OPC and URCNA denominations recently joint-published the Trinity Psalter-Hymnal, again without making it the official book for these denominations. Many URC have continued to use the 1958/76 CRC (their parent denomination) Psalter Hymnal since they broke away in 1996, generally preferring the older book to the CRC's 1987 update. The CRC maintained the copyright, and continued to print the old book and sell it while demand continued. I suspect there may be no future printings, and the URC intends to market the new PH on which they hold the (joint) copyright.
Many Presbyterian and Reformed churches in the USA also pay an annual license fee (church-size dependent) that allows them to use songs/lyrics (that may then be printed in a bulletin or projected on a screen/wall). These churches may have and use a standard hymnal (such as the TH or TPH or some other brand), but also desire to sing some recently published and popular songs. Again, this practice does not violate a denominational norm.
I hope this is helpful.