OK, I finally found some instructions that looked halfway trustworthy.

https://java3d.dev.java.net/binary-builds.html

has a readme for 1.5.1 that had relatively easy to follow
instructions. These actually worked! I've been able to view the little
"capeman" in a "java applet window" (I guess it doesn't do inlining in
the web page yet?)

In order to save people the trouble next time, I'll put an easily
found search string here:

In order to get Java3d working with sage in your browser, you have to
install the Java3d extension that can be found on "https://
java3d.dev.java.net/binary-builds.html". Follow the instructions in
the README posted there to install the Java3d extension in your local
java installation.

After that, the following actually works:

sage: from sage.plot.plot3d.shapes import *
sage: from sage.plot.plot3d.plot3d import plot3d
sage: S = Sphere(.5, color='yellow')
sage: S += Cone(.5, .5, color='red').translate(0,0,.3)
sage: S += Sphere(.1, color='white').translate(.45,-.1,.15) + Sphere(.
05, color='black').translate(.51,-.1,.17)
sage: S += Sphere(.1, color='white').translate(.45, .1,.15) + Sphere(.
05, color='black').translate(.51, .1,.17)
sage: S += Sphere(.1, color='yellow').translate(.5, 0, -.2)
sage: def f(x,y): return math.exp(x/5)*math.cos(y)
sage: P = plot3d(f,(-5,5),(-5,5), ['red','yellow'], max_depth=10)
sage: cape_man = P.scale(.2)+S.translate(1,0,0)
sage: cape_man.show()
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