Yes and no.
Johnson did not start US involvement in Vietnam. Direct American involvement started in 1954. That was during the administration of Dwight D. Eisenhower. It was in that year that the French left Vietnam and the US got more involved. By 1955, US advisers were in the country.
However, Johnson did bring on all-out American involvement in the war. It was in 1964 that the Gulf of Tonkin Resolution was passed and hundreds of thousands of US military personnel went to participate in the war. Therefore, it is wrong to say that Johnson started US involvement but it is correct to say that the US became fully involved in the war under him.