How do I make a double dabble circuit with logic gates
Any computable function can be implemented with enough logic gates if their implementation permits looping outputs back to inputs to construct flip flops, as most real ones do.

However, the key thing to realize is that because your algorithm of interest is iterative it is stateful rather than combinatorial - it requires memory elements such as registers feeding and fed by the combinational logic gates.

This would generally make implementing it with only logic gates (some in loop connection to create flip-flops) inadvisable; you probably want a mixture of logic gates and compatible-logic-family register ICs.

And more realistically there is no reason other than hobby enjoyment to build this with discrete parts at all.  You could for educational purposes build it in a programmable logic device (FPGA, or simply a simulator).

But because it is driving a display, ie, something that operates at only human speed, the usual "needs to be blindingly fast and/or parallel" justification for using dedicated logic rather than a small microcontroller isn't really there.   So pretty much any modern, "real problem" implementation would do this in software.