I will recommend you to apply through the Flipkart app if you want higher approval chances and are comfortable to pay ₹500. I also do not hold an axis bank account, and get rejected every time I apply through Axis Bank's website. But when I applied for the Flipkart Axis Bank card through the app, my card got approved within hours of doing video KYC. No axis bank agent visited my home and no income proof was asked from me. Fortunately, I applied from the app when it was offering the card as LTF. This LTF offer is a boon to me since I have got a meagre limit of ₹15000(I have heard that many people are getting only ₹15000 limit when applying from the app). This is the lowest limit card in my portfolio(which also contains cards with limit of nearly 2 lakhs). How can one expect to get the annual fee waiver by spending ₹2 lakh when he/she gets only 15k as the card limit? Anyways, I am rarely using this card for general transactions. I am waiting to get a limit increase offer. Meanwhile, I am extracting my card's value by mainly using it to access airport lounges.
So in short, applying through Flipkart app with get you higher chances of approval and no income docs requirement, but with the risk of getting a low limit and ₹500 fees. I think it is better than the lower approval chances on applying through Axis bank's website, and upon rejection, waiting for 90 days to file the next card application. I will recommend that before applying through any of the 2 mentioned methods, try to apply for the card through Bank Branch using card to card method.
P.S.: I am giving you all this advice as a student who does not have a income proof. My father files a decent ITR and he got his Axis Ace credit card approved without any hiccups and bank account by applying through Axis Bank's website.