The usual limit of emails is around 200-300 per hour with most hosts, a VPS server will allow you to handle the amount of emails you need to send. You have to read your host's AUP, as every company puts it in a different way. serverpoint.com states that
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We do not allow clients to send more than 1000 pieces of e-mail per day from a shared hosting account.
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Ahosting.biz indicates this in the resource Abuse Policy that the number of emails sent from the account depends on the number of domains hosted
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AVERAGE SERVER RESOURCE LIMITS
50 emails per hour, per domain
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To send 200K emails a day you will need a VPS server and provide verification that your emails are solicited.