You can create a ListedColormap from a list of colors. The following approach reads the image from the colormap and extracts the 45 colors of the centers of the colored rectangles. Note that the quality would be a bit better if the original image were saved ax png without passing through jpeg. As jpeg mixes the colors a bit, they might be slightly different from the originals.
import matplotlib.pyplot as plt
import numpy as np
img = plt.imread('desired_colormap.jpg')
# x and y positions of the rectangle centers are found via an image editor
xpos = 12
ypos_124 = 768
ypos_80 = 22
val_min = -124
val_max = -80
# extract the color from each of 45 rectangles
colors = [img[int(ypos), xpos] / 255
for ypos in np.linspace(ypos_124, ypos_80, val_max - val_min + 1)]
You can then use this list of colors to create a colormap and use it in seaborn. It looks a bit psychedelic.
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap
import seaborn as sns
import numpy as np
from scipy.ndimage import gaussian_filter
# colors were read from image
# create some smooth test data, with the given range of values
data = gaussian_filter(np.random.rand(1000, 1000), 20)
data -= data.min()
data *= (val_max - val_min) / data.max()
data += val_min
fig, ax = plt.subplots(figsize=(12, 10))
sns.set_style('white')
sns.heatmap(data=data, xticklabels=[], yticklabels=[],
cmap=ListedColormap(colors), vmin=val_min - 0.5, vmax=val_max + 0.5,
cbar_kws={'ticks': np.arange(val_min, val_max + 1)}, ax=ax)
plt.tight_layout()
plt.show()

PS: If you want to save the list for multiple use, you can convert the colors to hex (matplotlib.color.to_hex). The list would then be
colors = ['#000000', '#000000', '#68696b', '#00008c', '#0001cd', '#0102fa', '#4467e5', '#6296ea', '#4683b2',
'#4981b4', '#007f85', '#2d8a5d', '#619ca2', '#65cdaa', '#0afafb', '#02f99e', '#74fc02', '#adff31',
'#f6fbd2', '#f0e7ac', '#f1e68a', '#fbff0a', '#fdda00', '#dba51e', '#f9a608', '#ff8a06', '#d16920',
'#8d4312', '#fe804e', '#fe5945', '#fd4900', '#fe0300', '#fcb2c1', '#f0807c', '#fd6ba9', '#fc02f9',
'#dd9ee4', '#ba55d7', '#9202d6', '#85038b', '#7f017c', '#480181', '#4c0084', '#4b007f', '#480185']