I will assume by your problem statement and your limited types of tools, that you have limited woodworking experience. This means that any advice you receive should emphasize safety over expediency.
The safest way to cut out that shape from a solid or glued up wood section is with your jigsaw. First, a "master" template can be used to draw the shape on the wood and locate it for maximum use of clear, knot free material.
Second, after the shapes are cutout to within 1/16" of your line. they can be sanded, or hand planed to the exact dimension OR if you are comfortable using the plunge router with a template or pattern bit, that would reduce the sanding time immensely. Bits like these:
In this manner:
The pattern must be as smooth as possible since the resultant piece will be an exact duplicate. Use sticky tape sparingly to hold the pieces together while routing. and secure the workpiece to the bench top as well. This is important because you do NOT want anything to shift during the routing process!
The next issue I see is making the lap joints. A Japanese pull saw or a back saw will give you the control and fine cuts you need for accuracy. A sharp hand chisel and a small rabbet plane will be required to level out the bottoms of the laps. A word of Caution here. More folks have been severely cut using sharp hand tools than by power tools. I know this from my own experience! ..... power tools "0", ......hand tools "too many" to count or remember.
I would make the chairs one at a time, to make certain the design and the pieces all work together precisely. A whole batch of parts that are "wrong" won't make you a happy woodworker! Dry fitting each parts is the standard operating method to assure good fitting joints.
Clamping and gluing is the next step in the process. Strap clamps and squeeze clamps are handy when working with taper shapes and by yourself. A helper is recommended if you have one. Glue the pieces together in sub-assemblies rather than working your way around the chair from leg to leg. The sub assemblies would be the back and cross pieces, the front with the cross piece, and finally attach them together with the cross pieces.
Finishing would be the final step. I'll leave that for a later discussion.