More than likely yes. Why would someone do it free? Whether it is written or oral makes little difference, the valuer still put in the same amount of effort. If there was some kind of barter involved or family friend then maybe not. Depends on your discussions with the valuer.
I assume that you've obtained a house contents valuation because there is (or there is anticipated to be) disagreement in relation to the valuation of the house contents?
An independent valuer is generally unlikely to simply provide a verbal valuation/number that you can rely on in your family law proceeding.
If a valuation is performed by an independent valuer and both parties ultimately agree on the independent valuer's valuation, then its reasonable to request 50/50 split of the valuer's fee.
In contrast, my understanding is that if the valuation is (or is likely to be) disputed, then it is generally accepted that each party is liable for their own valuer's fees (as with any other expert fees) unless the parties or Court agree otherwise.
Hope this helps. Please keep us updated with your progress.