Is the sentence grammatically correct? Yes
Does it look odd and could it detract from your intended message? Yes
In many circumstances, you will choose to avoid an expression that follows all the conventions but is likely to draw attention away the meaning you intend to convey.
If one potential customer believe that there is a "only one and" rule and rejects your message because of that; then your campaign has failed.
The adjustment (not a correction, because what you have is not "wrong") can be as simple as replacing one of the uses of "and" with an ampersand.
Serious Savings on New & Used Motorhomes and Caravans
In the subject line of an email such advertising shorthand will not look out of place. In the body of the message, you could write "new or used caravans and motorhomes", without changing the (real world) meaning.