Correction, I do have to use both SetDefaultCurrency and SetDefaultCurrency, or else the currency won't switch when having no current order.

So the full working code, for anyone else who might need something similar, is this:

using System;
using System.Web;
using Umbraco.Core.Composing;
using Umbraco.Web.Routing;
using Vendlet.Library.Extensions;
using Vendr.Core;
using Vendr.Core.Services;
using Vendr.Core.Session;

namespace Vendlet.Library.Composers
{
    class PublishedRequestComposer : ComponentComposer<PublishedRequestComponent>
    {

    }

    public class PublishedRequestComponent : IComponent
    {
        private readonly ISessionManager _sessionManager;
        private readonly IUnitOfWorkProvider _uowProvider;
        private readonly IOrderService _orderService;
        private readonly HttpContextBase _httpContextBase;

        public PublishedRequestComponent(ISessionManager sessionManager,
            IUnitOfWorkProvider uowProvider,
            IOrderService orderService,
            HttpContextBase httpContextBase)
        {
            _sessionManager = sessionManager;
            _uowProvider = uowProvider;
            _orderService = orderService;
            _httpContextBase = httpContextBase;
        }

        public void Initialize()
        {
            PublishedRequest.Prepared += PublishedRequest_Prepared;
        }

        private void PublishedRequest_Prepared(object sender, EventArgs e)
        {
            var request = sender as PublishedRequest;

            HttpCookie language = _httpContextBase.Request.Cookies["language"];
            var currentCulture = request.Culture.TwoLetterISOLanguageName;

            //If the culture has been changed
            if (language == null || language.Value != currentCulture)
            {
                var content = request.PublishedContent;

                //Use PublishedContentExtension to get store
                var store = content.GetStore();

                //Use PublishedContentExtension to get root node
                var site = content.GetSite();

                //Vendr Store EntityPicket set to Currency
                var currency = site.Currency;

                //Vendr Store EntityPicket set to Country
                var country = site.Country;

                // applyToCurrentOrder does not seem to work, so we need to change currency on the current order, or else we get an error
                var currentOrder = _sessionManager.GetCurrentOrder(store.Id);
                if (currentOrder != null)
                {
                    using (var uow = _uowProvider.Create())
                    {
                        var order = currentOrder.AsWritable(uow).SetCurrency(currency);

                        _orderService.SaveOrder(order);

                        uow.Complete();
                    }
                }

                //Set default, will be used when there is no current order
                _sessionManager.SetDefaultPaymentCountry(store.Id, country.Id, applyToCurrentOrder: true);
                _sessionManager.SetDefaultCurrency(store.Id, currency.Id, applyToCurrentOrder: true);

                //Set cookie value to new language
                language = new HttpCookie("language");
                language.Value = currentCulture;
                _httpContextBase.Response.SetCookie(language);
            }
        }

        public void Terminate()
        {
            //unsubscribe during shutdown
            PublishedRequest.Prepared -= PublishedRequest_Prepared;
        }
    }
}