You haven't specified whether to assign a grey color scheme for your entire plot, or only for your lines. But just to not make things easy for myself, I'm going to assume the former. In that case, I would:
- use
template = 'plotly_white' for the figure elements not directly connected to your dataset, and
- assign a grey scale to all lines using
n_colors(lowcolor, highcolor, n_colors, colortype='tuple').
Example plot:

But as @S3DEV mentions, using the greys color palette could be a way to go too, and this is accesible through:
# In:
px.colors.sequential.Greys
# Out:
# ['rgb(255,255,255)',
# 'rgb(240,240,240)',
# 'rgb(217,217,217)',
# 'rgb(189,189,189)',
# 'rgb(150,150,150)',
# 'rgb(115,115,115)',
# 'rgb(82,82,82)',
# 'rgb(37,37,37)',
# 'rgb(0,0,0)']
And this would work perfectly for your use case with a limited number of lines. In that case you could just use this setup:
from plotly import offline as plot, subplots as subplot, graph_objects as go
from itertools import cycle
fig = subplot.make_subplots(rows=2, cols=1, shared_xaxes=True, vertical_spacing=0.01)
trace1 = go.Scatter(x = [1, 2, 3], y = [1, 2, 3])
trace2 = go.Scatter(x = [1, 2, 3], y = [4, 5, 6])
fig.append_trace(trace1, 1, 1)
fig.append_trace(trace2, 2, 1)
colors = cycle(list(set(px.colors.sequential.Greys)))
f = fig.full_figure_for_development(warn=False)
for d in fig.data:
d.line.color = next(colors)
fig.show()
And get:

And I assume that this is what you were looking for. But one considerable drawback here is that the number of colors in px.colors.sequential.Greys is limited, and I had to use a cycle to assign the line colors of your data. And n_colors(lowcolor, highcolor, n_colors, colortype='tuple') lets you define a starting color, an end color, and a number of colors scaled between them to form a complete scale for all your lines. This will also let you adjust the brightness of the colors to your liking. So you could get this:

...this:

or this:

Here's a complete setup for those figures if you would like to experiment with that as well:
import numpy as np
import pandas as pd
import plotly.graph_objects as go
import plotly.express as px
import datetime
from plotly.colors import n_colors
pd.set_option('display.max_rows', None)
pd.options.plotting.backend = "plotly"
# data sample
nperiods = 200
np.random.seed(123)
cols = 'abcdefghijkl'
df = pd.DataFrame(np.random.randint(-10, 12, size=(nperiods, len(cols))),
columns=list(cols))
datelist = pd.date_range(datetime.datetime(2020, 1, 1).strftime('%Y-%m-%d'),periods=nperiods).tolist()
df['dates'] = datelist
df = df.set_index(['dates'])
df.index = pd.to_datetime(df.index)
df.iloc[0] =1000
df = df.cumsum()#.reset_index()
greys_all = n_colors('rgb(0, 0, 0)', 'rgb(255, 255, 255)', len(cols)+1, colortype='rgb')
greys_dark = n_colors('rgb(0, 0, 0)', 'rgb(200, 200, 200)', len(cols)+1, colortype='rgb')
greys_light = n_colors('rgb(200, 200, 200)', 'rgb(255, 255, 255)', len(cols)+1, colortype='rgb')
greys = n_colors('rgb(100, 100, 100)', 'rgb(255, 255, 255)', len(cols)+1, colortype='rgb')
fig = df.plot(title = 'Greys_light', template='plotly_white', color_discrete_sequence=greys_light)
fig.update_layout(template='plotly_white')
fig.show()