The danger of salmonella and other infections mainly comes from eggs from large-scale industrialised farms where cramped conditions, poor nutrition and simply the large scale of production means a lot of chickens are vulnerable to infection and that it is possible for some eggs to be contaminated. If you are sourcing pastured eggs from small farms, they should be safe to use raw, especially if you wash them before cracking, since bacteria tend to reside on the eggshell.Just a point, on an aside.
However, if you are uncomfortable using raw eggs (or if you have immune issues/are pregnant in which case its better to be safe than sorry) - You could do it in a microwave, if you are not averse to that; I suppose this is the easiest pasturisation method I've seen: http://culinaryarts.about.com/od/eggsdairy/ht/pasteurize_eggs.htm
Traditional pasturization does take time and patience unfortunately. I think you can also buy pasteurised eggs at some supermarkets, though it's hard to vouch for their source;
pasturized egg whites are indeed easier to find, but keep in mind that raw egg whites contain avidin, and that the heat from the pasturisation process is not sufficient to break avidin bonds since you do not actually denature the protein. Some people have trouble digesting raw egg whites and, besides, avidin interferes with biotin absorbtion (its effectively an anti-nuitrient), so I wouldn't recommend it, and definitely not for frequent consumption. The yolk really contains the nutrition in the egg anyhow.
Nevertheless, here is a recipe:
1 cup egg whites
5 teaspoons salt
2 1/2 teaspoons mustard
1/2 cup lemon juice
about 2 cups mild oil (macadamia, avocado or melted ghee are best - because whites are less rich, something peppery like olive oil will give the mayo a very pronounced flavour.)
Place all ingredients except the oil in a blender/under a mechanic whisk and slowly add the oil as it thickens.
As for egg-free mayo, here is a recipe I found, if you do dairy:
4 tbsp unsweetened condensed milk
2 tsp Dijon mustard
salt and freshly ground black pepper
125ml oil of choice
2 tbsp cider vinegar
a squeeze of lemon juice
2 tsp full-fat greek yogurt
Blend until thickened.
Here is a dairy-free recipe:
1/2 cup nuts - cashews work best, macadamias also, possibly brazils - mild, buttery flavour is key
1/4 cup water
3 Tbsp vinegar or lemon juice
2 Tbsp oil of choice
1 1/2 tsp mustard
minced fresh/dried parsley and chopped chives (optional)
Grind the nuts to a flour consistency, then add the other ingredients until blended; transfer to a pot, and cook over medium-low until thick. Add more nuts to thicken if needed/more water to loosen.
Hope I helped!