[Solved]-List Inside IQueryable Object-LINQ,C#

May you could try something like this

var people = (from p in context.People
                   where p.Name == search.PersonName
                   select new Person{
                          ID = p.ID,
                        Name = p.Name,
                        Pets = (from pt in context.Pets
                                where pt.OwnerId == p.ID
                                select new Pet {
                                   id = pt.ID,
                                 OwnerId = pt.OwnerId,
                                 Name = pt.Name
                                })
                   }).Include("Pets").AsQueryable();

Or you could try something like this

foreach(var str in search.PetNames)
       {
            people.Concat(people.Where(o=>o.Pets.Any(p=>p == str)).Include('Pet'));
       }

Add AsQueryable() to Pets collection:

Pets = (from pt in context.Pets  
        where pt.OwnerId == p.ID  
        select new Pet {  
            id = pt.ID,  
            OwnerId = pt.OwnerId,  
            Name = pt.Name  
        }).AsQuerable() 

If Person and Pet are entities of your model and if Person.Pets is a navigation property to the Pet entity and if you want to have the full Person entity with all the full Pet entities and refering to your comment...

my method is supposed to return a list of people that have name equals search.PersonName & ALL their pets but only those people who own the pets with those names in the search.PetNames

...you could use this:

public List<Person> GetWithPets(SearchCriteria search)
{
    var people = from p in context.People.Include("Pets")
                 where p.Name == search.PersonName
                    && p.Pets.Any(pt => search.PetNames.Contains(pt.Name))
                 select p;

    return people.ToList();
}