What do you mean by synthesis? Articles and research? Systematic, qualitative methods? What you’re proposing has been done but is far less common than participants. Still, not all research uses participants, and the purpose and aims should drive your research. Yes, it is possible.

I question whether any part of a design that does not collect data can be called mixed methods. For example, in an exploratory sequential design, how would this be different from simply doing a literature review prior to designing a questionnaire or intervention?

Terminology is fickle, and not everyone approaches it the same way (and nomenclature seems to always be in flux). Many researchers embrace synthesis--this points to my thoughts that it depends on what the researcher is doing. Collecting and analyzing data from a literature review using qualitative methods can be quite different than simply writing up a literature review. For example, Onwuegbuzie, A. J., Leech, N. L., & Collins, K. M. (2012). Qualitative analysis techniques for the review of the literature. Qualitative Report, 17, 56. Retrieved from https://nsuworks.nova.edu/tqr/

Here are some examples that have been highly cited and influential:

Heyvaert, M., Maes, B., & Onghena, P. (2013). Mixed methods research synthesis: definition, framework, and potential. Quality & Quantity, 47, 659-676.

Stern, Cindy, Lucylynn Lizarondo, Judith Carrier, Christina Godfrey, Kendra Rieger, Susan Salmond, João Apostolo, Pamela Kirkpatrick, and Heather Loveday. "Methodological guidance for the conduct of mixed methods systematic reviews." JBI Evidence Synthesis 18, no. 10 (2020): 2108-2118.

Sandelowski, M., Voils, C. I., Leeman, J., & Crandell, J. L. (2012). Mapping the mixed methods–mixed research synthesis terrain. Journal of Mixed Methods research, 6(4), 317-331.

David C. Coker I agree that doing a full scale qualitative synthesis can be demanding, but I still do not see it as the equivalent of collecting and analyzing data. Instead, I think it has much more in common with other forms of literature review.

But there obviously is room to differ on this.

David L Morgan I think people get hung up on names versus what are they trying to accomplish. For example, I think the Sandelowski article is too theoretical and too abstract. But, I know editors and reviewers like succinct names. Quite the conundrum.