Vimgolf is a game where users of the text editor Vim try to transform one given document into another given document in as few keystrokes as they can.
Here's a toy example: You have two operations: "delete character under cursor" and "go to end of line". Now, with the cursor starting at the beginning of the string, transform the string # remove hashtags # into the string remove hashtags. The solution here would be two "delete character" operations followed by "go to end of line" followed by two "delete character" operations.
Is there an efficient, deterministic way to come up with solutions to vimgolf-style problems given the initial string, target string, and set of allowable operations?
I am not sure what you are asking - but I assume that you are asking for general case, like developing some algorithm with receiving allowed operations as input. I also assume that it is not that you are interested in toy example you gave (though as harder problem I explain below can be solved in polynomial time, I expect your problem can be solved in reasonable time), so instead of working on your example, I will provide what I know about the general problem.
I think it may depend on specific strings you give, and which operations you allow. In some cases, it might be very difficult if you include some very complicated vim macros. The number of possible solutions might be so big, so I am pretty confident that unless the operations are 'easy enough', there is no efficient way to solve general vimgolf problem. Of course you can work with computer, by enumerating a lot of key combinations and performing some search algorithms, and in some cases you might be able to cut a lot of impossible routes so you achieve the solution pretty fast.
I have seen a very specific vimgolf problem (as you call) in Baltic Olympiad of Informatics, which has different set of possible operations with what you gave, but in some case simillar. The solution and proof is long and complicated, but as you might be interested, here is the link to problem and solution description.
Problem : https://boi2013.informatik-olympiade.de/wp-content/uploads/2013/04/vim-ENG.pdf
Solution : https://boi2013.informatik-olympiade.de/wp-content/uploads/2013/05/vim-spoiler.pdf
In this problem, you have three operations - delete character, move cursor one step backward, and jump to next occurence of given character. It seems similar but harder than your example, and the solution link I wrote above explains $\mathcal{O}(NA^2)$ Solution where $N$ is size of string, $A$ is number of alphabet you use.