I hope you are well today and thank you for your question.
You can achieve this by adding the below CSS code in the Custom CSS option of Ivory Search plugin settings on the path Ivory Search -> Settings -> Settings
.site-header-inner-wrap .is-form-style input.is-search-input {
border-radius: 15px;
padding: 7px 11px;
width: 300px;
max-width: 100%;
}
It seems that your theme header is not responsive therefore the search form does not display responsively on mobile.
Could you please contact the theme support to make your website header responsive?
Best regards,
Ivory Search Support