I don't know about bicycle tubes. But I think it would be easier to make the drip irrigation system out of old garden hoses instead. The drip connectors fit the diameter of the garden hoses and they can be attached to a faucet timer or hose bib(with anti siphon valve attached). It will easily become a soaker with small holes punctured along the length of the hose. Bicycle tubes may stretch a bit too much under pressure and I don't know what the diameter is but I think it is considerably bigger than 1/2 inch. No worries about toxicity using a garden hose.
You could even attach the hose to a rain barrel instead of a faucet as long as you are running the lines downhill. Pressure from the barrel will be much less than from the faucet so it will drip slower. Lifting the barrel higher will increase pressure and flow.
I have also seen people turn pvc tubing in to drip tubing by drilling small holes along the length of the pvc. Cap the end and attach a 3/4 inch female coupling to the end. Attach a hose to that end and then attach the other end to a faucet(put in an anti siphon valve between the faucet and the hose to protect the water supply). The only downside is that the pvc is rigid so you can only lay it out in a straight line and the shape is not easy to change.