My '26 Roadster starts fine when the switch is on magneto. However, when I turn the switch to battery the car dies. Any idea what the problem may be?

Odd - usually it's the other way ? Possible incorrect wiring on switch.

Sounds like you lost the DC supply to the coil box. Someone can probably provide you a wiring diagram that can be very helpful. If you have a DC meter or even a 6 volt test bulb, you can verify the supply to the coil box. The engine does not have to be running to check the DC to the coil box. Just put the switch in the Bat position and the 6 Volt DC should be at the coil box.

Start with basics. The most likely cause is a bad connection. Check the switch for continuity and be sure all the wiring connections are tight.

Here's how things are connected.

Willie's test first. Then Steve's. The bulb test will tell you: 1, no power to the box? bad connection before the box (wiring, switch). 2, power at the box terminal? The boxes connections are bad at or inside it.

Clarification - I have a modern coil installed so how do I test the voltage as in Willie's test? Also, I replaced the magneto horn with a battery horn. Took the battery horn positive wire directly to post 4. Should I move it to post #1 as per the diagram?

You are using the mag terminal for the coil so when it's in the off position there is no voltage to it. When the key is turned to Bat. there is no connection to the coil. Run it that way. You are not running the coil off magneto voltage anyway. When cold a modern 6 (or 12) volt coil starts out at battery voltage and as it warms up ether the internal or external resistor drops the voltage to something like 4 volts. The magneto put out up to 36 volts AC which is too much. Also the magneto voltage is not a steady supply it has ups and down.
No. 4 is the tail light, your horn would only work if the lights are on. Battery horn would be hooked up to No 1 per the diagram.

Reading between the lines, the car does not appear to use Ford coils but rather a distributor. Whoever wired it probably wired the hot side of the coil to the magneto side of the ignition switch..