The second sentence is more common because it is an approximation of sloppy speech. The technically correct statement is, 'I hope you will be well soon.' The verb 'get' is an American catchall used as a substitute for numerous other verbs. In that you are hoping the person will be well in the near future, the verb must be in the future tense. Hence, 'I hope you'll get well soon' is the correct version of the sentence.
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