Welcome to the forum. I just remembered a couple of eggless cakes in a wartime cookery book I have by Marguerite Patten. A couple of them use milk but I suppose there might be a dairyless/soyless milk you could use (almond milk maybe?) With the last one which is "milkless"

if you didn't want it to be chocolate, I suppose you could just leave out the cocoa - maybe replace with a whole teaspoon of vanilla rather than a "few drops". I haven't tried any of these myself by the way...
Eggless Sponge
3oz (185g) margarine
4oz (110g) sugar
1 tbsp golden syrup
6oz (175g) self raising flour
1/4 pint (150ml) milk
Preheat the oven to Gas Mark 5/190C. Line a swiss roll tray 12 x 8-9 inches (30.5 x 20-23cm) with well greased greaseproof paper. Cream the margarine, sugar and syrup. Fold in the flour with the milk until smooth. Bake in the tin for 15 minutes or until firm to the touch.
Variation: Divide the mixture between two 6 1/2 -7 inch (16.5-18cm) sandwich tins. Bake at Gas Mark 4/180C for 18-20 minutes. Cool for 5 mins then turn out onto a wire rack to cool thoroughly. Sandwich with jam.
Eggless Ginger Cake
6oz (175g) self raising flour
1/2 - 1 tsp ground ginger
2oz (50g) margarine
2oz (50g) sugar
6 tbsp milk
3/4 tsp bicarbonate of soda
2 tsp vinegar (I assume it's talking white wine vinegar here rather than malt)
Grease and flour a 7-8 inch (18-20cm) cake tin. Preheat the oven to Gas Mark 5/190C. Rub the margarine into the flour and ginger, then mix in the sugar and milk. Blend the bicarb with the vinegar then beat into the cake. Bake in the tin for 20 minutes. Serve when fresh.
Eggless Chocolate Buns
8oz (225g) self raising flour
1oz (25g) cocoa powder
3oz (75g) sugar
7 1/2 floz (225ml) hot water
1 tbsp golden syrup
1 tsp bicarbonate of soda
Few drops of vanilla essence
3oz (75g) margarine, melted
Preheat the oven to Gas Mark 4/180C. Grease 12-5 patty tins. Mix the flour, cocoa and sugar. Pour the water into a good sized container. Add the syrup and bicarb. Add this to the flour mixure, along with the vanilla and margarine. Spoon into the tins and bake for 15-20 minutes until form to the touch. Remove from the tins and cool on a wire rack.
I assume you could also bake this last one in two sandwich tins to make one big birthday cake, probs for about 30 minutes - you could always experiment. Let us know how it goes.x.