Theoretically, dropshipping works without any investment. But

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1. You would need some cash to buy the products before you actually receive the money from your clients (eg: you use paypal to receive the funds from your customers and paypal needs 2-3 days to send your money to your bank account; during this time it would be a good idea for you to order the product and have it shipped to your customer, especially if you will use a dropshipping source which needs like 2-3 weeks to have the products delivered - AliExpress for example; furthermore, most customers are expecting to receive their orders pretty fast today...something like 5 days at max). So you would need to create a buffer of money which you could use for purchasing the products from your source before you actually receive the funds from your customers.
2. You could start by dropshipping on ebay, bonanza, etsy etc...where you don't have to pay anything upfront. But the competition is very tough there and you won't get too many sales, especially in the beginning because you will not know which products are hot sellers and they have not already been uploaded on those sites by your competitors. So, the wise choice here would be to start your own website (you'll need some money for domain name, host, theme - if you use wordpress) and some money for promoting your website (the cheapest option would be social media but you could try google ads, too).
So, theoretically you don't need money for starting a dropshipping business. But, in practice, you will need some money to get it running.