I agree with the above. Would be more helpful for us to help you to know more about your e-liquid and gear setup and your settings.
I'm assuming you have asthma or COPD since you're using an inhaler. Changing from smoking to vaping may irritate your lungs just because it is "different".
Many new vapers complain of coughing when first starting to vape. They fail to recall when they first began to smoke they probably coughed a lot. They eventually got used to the smoke and didn't cough so much ... until they got chronic diseases like asthma, chronic bronchitis, or COPD.
Inhaling warm vapor is new to your body, and its reacting to it like when you first began smoking. I imagine your body will eventually adjust to vaping. However, since I'm not a physician, you should monitor your new symptoms and if need be consult your physician.
You may want to adjust your vaping technique. Vaping requires a slightly different technique than smoking.
A Word on Vaping Technique
Smoking vs Vaping Differences