The name Wallin is NOT rare in Sweden so dates are needed for your Axel, plus other information which might show up on a Swedish database such as occupation, destination when he left Sweden, when he left Sweden, and so forth.
There is a cd called Emibas which is about 75% complete. Emibas was made from registrations in the Swedish parish to go to a foreign country. I see 31 Axel Wallin hits on Emibas and 10 Axel Vallin hits on the same cd. What information can you provide to see if your Axel is one of the 41 possibilities or perhaps isn't listed or perhaps is a man who used Wallin after emigrating and not before emigrating? (Swedes LOVE to change names.) If you have a full birthdate, that is best, but any clues are better than just a name.
Emihamn is another Swedish database made from registrations with Swedish port police before leaving Sweden. (Not all Swedes left through a Swedish port.) I see these Axel Wallin/Vallin hits but if yours had Axel as a given name but not the first given name, searching for Axel won't find him. There is a bit more information about these but this is the index. (I have ancestors who lived in Chicago but their destinations on this cd were not Chicago. Have an open mind about those with different destinations. I can't guess since you provided no indication at all when Axel left Sweden or when he was born or any other clues. Future marriages would of course not be listed on Swedish databases so that is not helpful searching the records I have available to me.
WALLINAXEL8VARBERGN1887BOSTON WALLINAXEL21AVESTAW1888NEW YORK WALLINAXEL36A1890WORCESTER WALLINAXEL1SUNDSVALLY1893NEW YORK WALLINAXEL25WIBYE1894CHICAGO ILLS WALLINAXEL8A1895BOSTONMA WALLINAXEL29AMERIKAUS1906ROCKFORD IL WALLINAXEL39AMERIKAUS1907NEW YORK WALLINAXEL18RÄNGSJÖX1913MINNEAPOLIS WALLINAXEL39NYDALAF1921NEW YORK WALLINAXEL62AMERIKAUS1930NEW YORK WALLINAXEL18AMERIKAUS1930NEW YORK WALLINAXEL A24LERBÄCKT1901BOSTON MASS WALLINAXEL B24AMERIKAUS1909CHICAGO WALLINAXEL E19BJERTÅY1903MARINETTE VI WALLINAXEL G19TÖREBD1929NEW YORK WALLINAXEL H31STJERNARPE1903NEW YORK WALLINAXEL L23FÄSSBERGO1891OMAHA WALLINAXEL R20STOCKHOLMA1905GRAND RAPIDS AXEL10ATTMARY1893RAT PORTAGE VALLINAXEL25VIBYE1894CHICAGO ILLS VALLINAXEL31AMERIKAUS1926NEW YORK VALLINAXEL R24AMERIKAUS1908NEW YORK NY
I wrote a message about this family to Lynette back in 2004. This family does have an Axel but Axel is a very common name.
Many researching Swedish ancestry use the Swenson Swedish Immigration Research Center. The Swenson Center has a website but its extensive collection of records is NOT online. We need to go there to research or pay them to research for us. They have records which are difficult or even impossible to find elsewhere and many tough problems are solved using this archive.